In a food processor or bowl, mix 2 cups flour, 1/4 cup sugar, baking powder, and salt.
Cut the 3/4 cup butter into small pieces and add to flour mixture. Whirl or cut in butter with a pastry blender until mixture has texture of fine crumbs.
Beat milk and egg to blend. Pour into flour mixture and whirl or stir with a fork just to evenly moisten. On a lightly floured board, knead dough briefly until it holds together.
With floured hands, pat the dough evenly over bottom of a 9- by 13-inch pan and press edges up sides to form a slight ridge.
Place apple slices, overlapping slightly, in 3 rows down the length of the dough.
In a food processor or bowl, whirl or rub with your fingers the remaining 2 tablespoons butter, 1 cup sugar, and cinnamon until mixture has texture of fine crumbs. Sprinkle over apples.
Bake in the middle of a 350° oven until cake is browned, about 40 minutes. Cut into squares. Serve cake warm or cool.