Preheat oven to 350°F (180°C). Spray a 9-inch square baking pan with cooking spray; line pan with parchment paper, letting excess extend over sides of pan.

In the bowl of a stand mixer fitted with the paddle attachment, beat butter, ⅔ cup (133 grams) granulated sugar, and brown sugar at medium speed until fluffy, 3 to 4 minutes, stopping to scrape sides of bowl. Add eggs, one at a time, beating well after each addition. Beat in vanilla bean paste.

In a medium bowl, whisk together flour, baking powder, salt, and ginger. Add flour mixture to butter mixture alternately with sour cream, beginning and ending with flour mixture, beating just until combined after each addition. Spoon half of batter (about 438 grams) into prepared pan; using a small offset spatula, smooth into an even layer.

In a medium bowl, stir together cardamom and remaining ⅓ cup (67 grams) granulated sugar; add 1¼ cups (224 grams) peaches, tossing until well combined. Spoon peach mixture evenly onto batter in pan. Spoon remaining batter onto peach mixture, smoothing into an even layer. (It’s OK if some peaches peek through.) Top with Oat Streusel and remaining 1¼ cups (224 grams) peaches.

Bake until a wooden pick inserted in center comes out clean, 50 minutes to 1 hour, rotating pan halfway through baking and loosely covering with foil to prevent excess browning, if necessary. Let cool in pan on a wire rack for 10 minutes. Run a sharp knife around sides of pan between paper and pan. Using excess parchment as handles, remove cake from pan. Garnish with confectioners’ sugar, if desired. Serve warm or at room temperature.