Preheat oven to 350°F (175°C). Grease a 9x13-inch baking pan. Make the crumble: stir flour, brown sugar, and cinnamon; cut in cold butter with fingertips until coarse crumbs form with some pea-sized pieces. Set aside.

Whisk flour, baking powder, baking soda, and salt in a medium bowl.

In a large bowl, beat butter and sugar with an electric mixer until light and fluffy, about 2 minutes. Beat in eggs one at a time, then sour cream and vanilla.

Gradually add dry ingredients to wet, mixing on low just until no dry streaks remain. Spread half the batter in the pan, top with half the peaches and half the crumble. Add remaining batter, then remaining peaches and crumble.

Bake 40 to 45 minutes, until golden and a toothpick in the center comes out with a few moist crumbs. Cool in the pan on a rack for 30 minutes. Dust with powdered sugar, slice, and serve.