Butter the cake pan. Set aside.
Make the topping: In a small bowl, stir together the brown sugar, flour, and cinnamon. Sprinkle the 3 tablespoons of butter cubes over the mixture and use your fingertips to work it in to form a crumbly topping. Chill in the refrigerator while you prepare the cake.
Preheat the oven to 350 degrees F. Make the cake: In a small pan, melt the butter over medium heat until it begins to turn a medium nut brown, 6 to 8 minutes. Remove from the heat and set aside to cool slightly. In a large bowl, whisk together the flour, sugars, baking powder, baking soda, salt, and spices. In a medium bowl, beat the egg and egg yolk to blend. Whisk in the butter and buttermilk, then pour the liquid mixture into the flour mixture and stir just to combine. Fold in the apple cubes and pour the batter into the prepared pan. Sprinkle the topping over the cake and arrange the apple slices around the top, lightly pressing them into the topping. Bake until the top is golden brown and a cake tester inserted into the center comes out clean, 45 to 55 minutes. Set on a cooling rack for 20 minutes, then check to see if the sides of the cake are sticking to the pan in any spots. If so, use a thin knife to loosen them. Carefully remove the sides of pan, transfer to a platter, and serve warm or at room temperature.
Recipe from The Apple Lover's Cookbook (c) 2011, by Amy Traverso. Reprinted with permission by W. W. Norton & Company.