Cut 1/2 cup butter into 1 cup flour with a knife or pastry blender in a bowl until the mixture resembles coarse crumbs. Stir in 1/4 teaspoon butter extract.
Add the water a tablespoon at a time, tossing with a fork, until the flour mixture is moistened. Press mixture onto the bottom of a 10x13-inch baking sheet.
Bring 1 cup water, 1/2 cup butter, and 1/2 teaspoon butter extract to a boil in a saucepan over medium-high heat.
Remove from heat, then beat in 1 cup flour. Beat in eggs one at a time until blended. Stir in 1 teaspoon almond extract, and pour over crust.
Bake in the preheated oven until golden brown, about 55 minutes.
Meanwhile, mix the confectioners' sugar, 1/2 teaspoon almond extract, 1/4 teaspoon butter extract, and 1 tablespoon milk in a bowl until smooth.
