On a lightly floured surface, roll dough to a 1/8-in.-thick circle; transfer to a 9-in. pie plate. Trim to 1/2 in. beyond rim of plate; flute edge. Refrigerate 30 minutes. Preheat oven to 425°.
Line unpricked crust with a double thickness of foil. Fill with pie weights, dried beans or uncooked rice. Bake on a lower oven rack until edge is light golden brown, 15-20 minutes. Remove foil and weights; bake until bottom is golden brown, 3-6 minutes longer. Cool on a wire rack. Reduce oven setting to 350°.
In a saucepan, melt butter. Remove from the heat; add flour and stir until smooth. Stir in brown sugar. Return to heat; stir in milk and salt until blended. Cook and stir over medium-high heat until thickened and bubbly. Reduce heat; cook and stir 2 minutes longer. Remove from the heat. Stir about 1 cup hot filling into egg yolks; return all to pan, stirring constantly. Bring to a gentle boil; cook and stir for 2 minutes longer. Remove from the heat. Gently stir in vanilla. Pour into crust.
For meringue, beat egg whites and cream of tartar in a small bowl on medium speed until soft peaks form. Gradually beat in sugar, about 1 tablespoon at a time, on high until stiff glossy peaks form and sugar is dissolved. Spread evenly over hot filling, sealing edge to crust.
Bake until meringue is golden brown, 12-15 minutes. Cool on a wire rack for 1 hour. Refrigerate at least 3 hours before serving. Refrigerate leftovers.
Combine 1-1/4 cups all-purpose flour and 1/4 tsp. salt; cut in 1/2 cup cold butter until crumbly. Gradually add 3-5 Tbsp. ice water, tossing with a fork until dough holds together when pressed. Shape into a disk; wrap and refrigerate 1 hour.