Preheat oven to 375°F (190°C). Spray a 12-cup muffin pan with cooking spray.

In the bowl of a stand mixer fitted with the paddle attachment, beat brown sugar, ¼ cup (50 grams) granulated sugar, and egg at medium speed until lightened in color and well combined, 1 to 2 minutes, stopping to scrape sides of bowl. Add ¼ cup (57 grams) melted butter, oil, and vanilla bean paste; beat until well combined, 1 to 2 minutes, stopping to scrape sides of bowl.

In a medium bowl, whisk together flour, salt, baking soda, nutmeg, and ¼ teaspoon cinnamon. With mixer on low speed, add flour mixture to sugar mixture alternately with buttermilk, beginning and ending with flour mixture, beating just until combined after each addition and stopping to scrape sides of bowl. Divide batter evenly among prepared cups (about ¼ cup [about 70 grams] each). Firmly tap pan on a kitchen towel-lined counter several times to settle batter and release any air bubbles.

Bake until a wooden pick inserted in center comes out clean, 12 to 18 minutes. Let cool in pan for 5 minutes. Remove from pan, and let cool completely on a wire rack.

In a small shallow bowl, place remaining ½ cup (113 grams) melted butter.

In a medium shallow bowl, stir together remaining ¾ cup (150 grams) granulated sugar and remaining 1½ teaspoons (3 grams) cinnamon. Roll a doughnut in melted butter; roll in cinnamon sugar. Repeat with remaining doughnuts. Serve immediately.