Heat oven to 350°F. Set the baked crust on a sheet pan.
Whisk sugar, cocoa, cornstarch, and salt in a medium saucepan.
Slowly whisk in the milk until smooth with no dry pockets.
Cook over medium heat, whisking constantly, until thick bubbles form, 5–8 minutes.
Temper the yolks: whisk yolks in a bowl, then slowly whisk in 1 cup hot mixture.
Return yolk mixture to the saucepan and cook 1 minute, whisking, until glossy and thick.
Remove from heat; stir in butter and vanilla until silky.
Pour the hot chocolate filling into the crust and smooth the top.
Beat egg whites and cream of tartar on medium-high to soft peaks.
Gradually rain in the sugar; beat to stiff, glossy peaks. Mix in vanilla just to combine.
Spread meringue over hot filling, sealing to the crust. Swirl the top.
Bake until meringue is golden, 12–15 minutes. Cool, then chill 2 hours before slicing.