Preheat the oven to 350°F (175°C). Grease and line a 9-inch square pan with parchment.

In a bowl, whisk together the flour, brown sugar, granulated sugar, cinnamon, and salt. Stir in the melted butter until clumps form. Fold in nuts if using and chill.

In another bowl, toss the diced apples with lemon juice, sugar, cinnamon, and nutmeg, and set aside.

In a separate bowl, whisk the flour, baking powder, baking soda, and salt.

In a large bowl, beat the softened butter with granulated and brown sugar until fluffy, about 1–2 minutes.

Add eggs one at a time followed by the vanilla extract.

Mix in half the dry ingredients, then add sour cream, followed by the remaining dry ingredients. Add milk 1 tablespoon at a time until batter is thick but smooth.

Spread about 2/3 of the batter into the pan. Scatter the apple mixture evenly, dollop with the remaining batter.

Sprinkle the chilled crumb topping over the batter. Bake for 45–55 minutes until a toothpick comes out clean or with a few moist crumbs.

Cool in the pan for 20 minutes, then lift out to a rack. Dust with powdered sugar or drizzle caramel before slicing.