In this adaptation of old-fashioned Shaker pie, thin slices of whole lemon are soaked with sugar overnight to soften and sweeten the peel. When mixed with eggs, they create a soft custard base with chewy lemon pieces and a pleasantly bitter edge.
Quarter lemons lengthwise. Slice quartered lemons crosswise as thinly as possible, discarding the end pieces that are just peel and pith. Put lemon slices in a large bowl with the sugar. Toss to combine, cover, and let sit at least 24 hours and up to 2 days at room temperature.
Put an oven rack on the lowest rung and preheat oven to 450°. Lightly dust counter and rolling pin with flour, then unwrap dough. With short strokes from center outward, roll dough into a 12-in. circle (about 1/8 in. thick), turning 90° after every 3 or 4 passes of the rolling pin to keep it from sticking. Transfer dough to a 9-in. pie pan, letting it fall into place (if you push or stretch the dough, it will shrink back when baked). Trim overhang to 1/2 in., then tuck edge under. Cover with plastic wrap and refrigerate 20 minutes.
Meanwhile, whisk the whole eggs and egg yolk in a medium bowl until frothy. Drain lemons (reserving liquid) and add lemon slices and 1 1/4 cup of the reserved liquid to the eggs. Stir to combine. Pour filling into crust. Cover with foil and bake 15 minutes. Remove foil, reduce heat to 325°, and bake another 20 minutes. Remove from oven, but leave oven on.
In a large clean bowl, beat egg whites, salt, and cream of tartar to create soft peaks. Add brown and granulated sugars and beat until shiny, firm peaks form. Spread meringue in a mound over hot pie and return to oven. Bake until top is puffed and deep brown, about 25 minutes. Let cool completely, at least 3 1/2 hours.