As Elva Miller butchers one pop hit, church hymn or acid rock song after another, in a voice that sounds like a shrieking cat with whooping cough, the question nags at a listener: Does she have the slightest inkling that she’s hideous?

That’s the enticing mystery ­animating “Mrs. Miller Does Her Thing,” James Lapine’s pleasantly offbeat new bio-play with music at Signature Theatre. And, in the guise of actress Debra Monk, cheerful cluelessness has never been quite so beguiling.

Who’s making that racket? None other than Debra Monk.

The seriocomedy, in its world premiere on Signature’s main stage, posits Mrs. Miller — a real-life show business novelty whose egregious LPs climbed the charts in the mid-to-late 1960s — as a touchstone of a national cultural shift. With a younger generation rising up against discrimination and a war in Asia, expressing its resistance in the music of the counterculture, an older, more conservative one struggled to preserve quickly calcifying norms of the American mainstream.

Mrs. Miller, Lapine asserts, was an emblem of the confusion at the intersection of these opposing forces. Plucked by irreverent (and perennially stoned) record producers from the obscurity of white-bread suburbia, she recorded tone-deaf covers of Top 40 hits by the Beatles, Johnny Mercer and Petula Clark. It made her a star, though one who burned out rapidly. Her seeming obliviousness to her own lack of talent is what did the trick: She whistled into the winds of change, blissfully unconcerned that they would soon blow her away.

The achievement in Lapine’s efficiently staged production revolves chiefly around the performance of the primly coifed and outfitted Monk, who nails the impersonation in a way that’s both funny and sad. Mrs. Miller is presented as the kookiest entertainer of her time, expressing a joy that transcends the ridiculousness of her chaotic renditions of “Downtown” and “I Want to Hold Your Hand.” The wackiest has her in a hard-rock phase, singing Jefferson Airplane’s “White Rabbit.” The screeching Monk over-enunciating the lyrics — “And you’ve just had some kind of mushroom, and your mind is moving low” — is hilariously far out. A four-member band led by Matt Hinkley supplies fine accompaniment for Monk and the sunny dancing backup trio of Kimberly Marable, Kaitlyn Davidson and Jacob ben Widmar.

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The task of imbuing this material with specialness presents challenges, one of them out of Lapine's control. The story bears strong similarities to another tale of ghastly feminine crooning that's still fresh in the public's mind — that of Florence Foster Jenkins, the subject of a 2016 movie by that name about a 1930s American socialite played by Meryl Streep who couldn't carry a tune and insisted on embarrassing herself in public. In that case, though, Jenkins's own financial support for a variety of organizations was the guarantee that she remained happily floating on a cushion of unearned compliments.

Mrs. Miller is the butt of a more obvious joke, in a less genteel time, when contempt for American institutions was growing and it was becoming ever cooler to mock anyone who wasn’t hip to the revolution. To that end, Lapine’s Mrs. Miller seems as strangely unaware of the pain in the world around her as she is of the truth about her own gifts. (At a few key moments, Monk’s singing segues into sweet melodiousness, indicating how it all sounds in her own head.)

That pain is embodied by her loyal if morose niece, Joelle (Rebekah Brockman); a failing young record producer (Corey Mach) facing military conscription; and, most sourly, by Elva’s husband, John (Boyd Gaines), a miserable old coot uttering anti-Semitic remarks as he lives out his days as a stroke victim in a nursing home. The actors don’t have enough to do beyond serving as depressed counterpoints to the eternally vivacious Mrs. Miller, and so their scenes come across monochromatically. The ’60s may have been psychedelic, at least as far as designer Jennifer Caprio’s snazzy costumes suggest, but here the decade is too frequently a downer.

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Except, of course, for the almost exasperatingly upbeat Mrs. Miller. By the time she’s been recruited for a USO tour with Bob Hope in South Vietnam — where she sings in a jungle-warfare outfit complete with hand-grenade belt and camouflage heels — we have grown a bit weary of her dizzy caterwauling. And our curiosity about how much she’s in on the joke is in danger of abating. Perhaps some refinement is needed here, so that an audience understands more incisively what drives Mrs. Miller and doesn’t overdose on the terrible vocalizing. What’s also missing is some sense of who the fans were who so ardently embraced her act.

Fortunately, Lapine and Monk do suggest a solution to the evening’s biggest riddle. It won’t be divulged here. But thankfully, we do leave “Mrs. Miller” with the sense that she may not be as much of a fool as Hollywood was happy to have the public believe.

Mrs. Miller Does Her Thing, written and directed by James Lapine. Choreography, Josh Prince; orchestrations, Michael Starobin; set, Heidi Ettinger; music direction, Matt Hinkley; costumes, Jennifer Caprio; lighting, Jeff Croiter; sound, Ryan Hickey; wigs and makeup, J. Jared Janas; production stage manager, Kerry Epstein. With Will LeBow. About 1 hour 45 minutes. Tickets: $40-$99. Through March 26 at Signature Theatre, 4200 Campbell Ave., Arlington. Visit sigtheatre.org or call 703-820-9771.

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