Place a baking sheet in the oven and preheat to 400 degrees F. Line a 9-inch pie plate with 1 piece of dough.

Lay the remaining piece of dough on a lightly floured surface and spread the butter evenly on top. Combine 1/4 cup granulated sugar and 1 teaspoon cinnamon in a small bowl; sprinkle evenly over the butter, then gently press with your fingers to help the mixture adhere.

Roll the pie dough into a tight log. Trim and discard about 1 1/2 inches from both ends; cut the remaining log crosswise into 1/2-inch-thick pieces.

Arrange the pieces cut-side down in a snug circle on floured parchment paper. Lightly dust with more flour, then gently roll out into a 10-inch round.

Peel and thinly slice the apples. Toss with the lemon juice, vanilla and the remaining 1/4 cup granulated sugar and 1/8 teaspoon cinnamon in a large bowl. Transfer to the dough-lined pie plate; invert the cinnamon-roll crust on top and peel off the parchment. Pinch the crusts together; fold the overhanging dough under itself and crimp as desired. Brush with the beaten egg.

Set the pie on the hot baking sheet in the oven; bake until the crust is golden brown and the filling is bubbling, about 50 minutes.

Whisk the confectioners' sugar and 2 tablespoons milk in a bowl until smooth. (Add more milk if the glaze is too thick.) Drizzle over the pie.