Arrange a rack in the middle of the oven and heat to 350°F (175°C). Grease a 9x13-inch pan or line with parchment, and bring the eggs and sour cream to room temperature.
Whisk 3/4 cup flour, brown sugar, 1 1/2 teaspoons cinnamon, and 1/4 teaspoon salt. Stir in the melted butter with a fork until clumps form, then set aside.
Peel, core, and dice the apples into 1/2-inch pieces. Toss with 1 tablespoon flour until lightly coated.
In a separate bowl, whisk 2 1/2 cups flour, baking powder, baking soda, 1/2 teaspoon salt, and 1 teaspoon cinnamon until evenly combined.
Beat the softened butter and granulated sugar until pale and fluffy, 2 to 3 minutes. Add the eggs one at a time, then mix in the vanilla and sour cream until smooth.
Add the dry ingredients to the wet in two additions, folding gently with a spatula just until no dry flour remains. The batter will be thick.
Fold in the floured apples, including any loose flour, just until evenly distributed.
Spread the batter evenly in the prepared pan. Squeeze the streusel into small clumps and scatter over the top, pressing down very lightly.
Bake at 350°F (175°C) for 40 to 45 minutes, until golden and a toothpick in the center comes out with only a few moist crumbs. Tent with foil if the streusel browns too fast.
Cool on a wire rack for at least 20 minutes. If glazing, whisk the powdered sugar, milk, and vanilla until smooth, drizzle over the warm cake, and let set before slicing into 12 squares.