Heat oven to 350°F. Grease and parchment-line a 9×13-inch pan.
Make the crumble: mix flour, oats, brown sugar, cinnamon, and salt. Cut in cold butter until clumpy. Fold in pecans.
Toss apples with sugar, cinnamon, lemon juice, and cornstarch. Set aside to juice slightly.
Cream butter and sugar until light and fluffy in a large bowl.
Beat in vanilla, then slowly stream in the beaten eggs until smooth.
Whisk flour, baking powder, baking soda, and salt. Stir into the bowl with sour cream just until combined.
Spread batter in the pan. Spoon apples evenly over the top. Cover with the crumble.
Bake 43–48 minutes, until a tester comes out with a few moist crumbs. Cool 20 minutes.
Whisk powdered sugar, milk, and vanilla until pourable. Drizzle over warm cake, slice, and serve.