Stir flour, baking powder, baking soda, salt, cinnamon and nutmeg and set aside.
Cream the butter and sugar on medium speed with the paddle attachment for 1 minute. Scrape and beat for an additional minute or until noticeable lighter in color.
Add eggs and beat until homogenous. Add yolks and beat until smooth. Scrape and beat for 30 seconds. Add 1/3 of the flour mixture and beat on low speed until just barely combined. Add in half the sour cream, mixing again until just combined. Repeat and end with the last 1/3 of the flour. Fold in the grated carrots.
Drop dough onto generously floured surface and let it rest for 15 minutes.
Dust the top of the dough with flour and roll out to ¾” thickness. Cut out doughnuts and holes and place onto floured baking pan (preferably lined with parchment paper) OR oil parchment paper squares.
Heat frying oil to 370F. Fry 90 seconds on one side. Flip and fry for 1 minute on the other. Remove from heat and place on cooling rack. Glaze while just barely warm.
Glaze
Place all ingredients in a food processor and blend until smooth. Just before glazing, stir in 1 tsp very hot water to loosen the glaze.