Combine almond flour, sugar, all-purpose flour, and salt in a small bowl. Work in butter with your fingers until mixture holds together when pinched but is still crumbly; cover and chill.
Cook chestnuts, 2 Tbsp. brown sugar, and 2 Tbsp. water in a small skillet over medium heat, stirring constantly, until pan is almost dry and chestnuts are coated in syrup, about 5 minutes. Transfer to a small bowl and let cool. Toss in cocoa and remaining 1 Tbsp. brown sugar.
Preheat oven to 350°. Butter pan and line bottom with a round of parchment paper; butter parchment and flour pan. Whisk baking powder, baking soda, salt, and 2 cups flour in a medium bowl.
Using an electric mixer on high speed, beat sugar and ½ cup butter until light and fluffy, about 4 minutes. Add eggs one at a time, beating to blend between additions. Beat in sour cream and vanilla. Mix in dry ingredients. Scrape half of batter into prepared pan. Sprinkle chestnut filling over; top with remaining batter, then streusel.
Place pan on a rimmed baking sheet and bake cake until top is golden brown and a cake tester inserted into the center comes out clean, 45–55 minutes. Transfer pan to a wire rack and let cake cool before unmolding.
DO AHEAD: Cake can be baked 1 day ahead. Store tightly wrapped at room temperature.
