Place yeast in the bowl of a mixer. Pour milk over yeast; stir to combine. Let stand until foamy, about 2 minutes. Add 2 3/4 cups flour to bowl without stirring. Cover tightly with plastic, and let stand in a warm place until surface is cracking, about 45 minutes.
Whisk together cup flour, the yolks, brown and granulated sugars, vanilla, salt, and cinnamon.
Melt butter in a small saucepan over medium heat, swirling occasionally, until dark brown with a nutty aroma. Remove from heat. Pour browned butter (with browned bits) over yolk mixture; whisk to combine.
Add yolk-butter mixture to yeast mixture. Beat with a mixer on medium-low speed until just combined. Add remaining 1/2 cup flour. Beat on medium-high speed until combined, about 2 minutes (dough will be sticky). Transfer to an oiled bowl; cover. Let stand in a warm place until doubled in volume, about 2 hours. Turn out dough onto a floured surface, and knead for 2 minutes. Return dough to bowl. Let stand until doubled in volume, about 30 minutes more.
Turn out dough onto a lightly floured surface. Gently roll out dough into a 10-by-25-inch rectangle. Cut crosswise into twenty 1/2-inch-thick strips. Bring ends of 1 strip together, twist, and pinch ends to seal. Place on a baking sheet. Repeat with remaining strips. Let stand for 15 minutes.
Heat 2 to 3 inches oil in a heavy 5-quart saucepan until a deep-fry thermometer registers 375 degrees. Working in batches, fry doughnuts, turning occasionally, until browned, about 2 minutes. (Adjust heat as necessary to maintain oil temperature.) Transfer doughnuts to paper-towel-lined plates to drain; let cool slightly. Toss doughnuts in superfine sugar. Serve with chocolate-brandy sauce if desired.