Adjust oven rack to middle position and heat oven to 425 degrees. Line rimmed baking sheet with aluminum foil and spray with vegetable oil spray. Gently toss 24 peach wedges with 2 tablespoons schnapps, 2 teaspoons lemon juice, and 1 tablespoon granulated sugar in bowl; set aside.
Cut remaining peach wedges crosswise into thirds. Gently toss chunks with remaining 3 tablespoons schnapps, remaining 2 teaspoons lemon juice, and 2 tablespoons granulated sugar in bowl. Spread peach chunks in single layer on prepared sheet and bake until exuded juices begin to thicken and caramelize at edges of sheet, 20 to 25 minutes. Transfer sheet to wire rack and let peaches cool to room temperature, about 30 minutes. Reduce oven temperature to 350 degrees.
Spray 9-inch springform pan with vegetable oil spray. Whisk flour, baking powder, and salt together in bowl. Whisk brown sugar, 1/3 cup granulated sugar, and eggs together in second bowl until thick and homogeneous, about 45 seconds. Slowly whisk in butter until combined. Add sour cream, vanilla, and 1/4 teaspoon almond extract; whisk until combined. Add flour mixture and whisk until just combined.
Transfer half of batter to prepared pan; using offset spatula, spread batter evenly to pan edges and smooth surface. Sprinkle crushed bread crumbs evenly over cooled peach chunks and gently toss to coat. Arrange peach chunks on batter in even layer, gently pressing peaches into batter. Gently spread remaining batter over peach chunks and smooth top. Arrange reserved peach wedges, slightly overlapped, in ring over surface of cake, placing smaller wedges in center. Stir together remaining 3 tablespoons granulated sugar and remaining 1/8 teaspoon almond extract in small bowl until sugar is moistened. Sprinkle sugar mixture evenly over top of cake.
Bake until center of cake is set and toothpick inserted in center comes out clean, 50 to 60 minutes. Transfer pan to wire rack; cool 5 minutes. Run paring knife around sides of cake to loosen. Remove cake from pan and let cool completely, 2 to 3 hours. Cut into wedges and serve.
**TIP: Keeping peaches in the fridge might seem like a good way to prolong their shelf life, but unless the fruit is ripe, the cold temperatures can turn their flesh mealy. Storing the fruit at or below 40 degrees deactivates an enzyme that breaks down its pectin during ripening. If this happens before the flesh is ripe, the pectin will remain intact and the flesh texture will be mealy. The lesson: Store peaches on the counter.