Preheat oven to 300°F (150°C). Spray an 8-inch square baking pan with baking spray with flour. Line pan with parchment paper, letting excess extend over all sides of pan.

In the bowl of a stand mixer fitted with the paddle attachment, beat butter, granulated sugar, and brown sugar at medium speed until light and fluffy, 3 to 4 minutes, stopping to scrape sides of bowl. Add eggs, one at a time, beating until combined after each addition and stopping to scrape bottom and sides of bowl. Beat in vanilla.

In a medium bowl, whisk together flour, baking powder, and salt until combined. In a small bowl, whisk together sour cream and cottage cheese. With mixer on low speed, gradually add flour mixture to butter mixture alternately with sour cream mixture, beginning and ending with flour mixture, beating until just combined after each addition and stopping to scrape bottom and sides of bowl. Add chocolate chips, beating just until combined. Spread batter into prepared pan.

Bake until a wooden pick inserted in center comes out with a few moist crumbs and an instant-read thermometer inserted in center registers 205°F (96°C), 1 hour and 10 minutes to 1 hour and 20 minutes, covering with foil during final 20 minutes of baking to prevent excess browning. Let cool in pan for 10 minutes. Using excess parchment as handles, remove from pan, and let cool completely on a wire rack.

Brush melted butter on top of cooled cake; sprinkle with granulated sugar and chocolate chips. Store in an airtight container for up to 3 days.