Place brown sugar, granulated sugar, and butter in a large bowl; beat with a mixer at medium speed until well blended (about 3 minutes). Add egg, beating well. Beat in 1/2 cup sour cream and vanilla. Weigh or lightly spoon flour into dry measuring cups; level with a knife. Combine flour, baking powder, salt, nutmeg, and cinnamon, stirring well with a whisk. Add flour mixture to sugar mixture; beat at low speed just until combined.
Spoon batter into a large zip-top plastic bag. Snip a small hole in 1 bottom corner of bag; pipe batter evenly into 10 doughnut cups coated with baking spray, filling about two-thirds full. Bake at 375° for 14 to 15 minutes or until a wooden pick inserted in center comes out clean. Cool on a wire rack.
Combine powdered sugar and remaining 1 tablespoon sour cream in a small bowl, stirring with a whisk until smooth. Drizzle evenly over doughnuts. Sprinkle evenly with nuts.