Paczki (Authentic Polish Doughnuts for Fat Tuesday aka Paczki Day)

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1. 1Step 1: Make sure all ingredients are room temperature. Really. Leave them out overnight at least. You can even soften the butter some more by microwaving a mug of water and leaving the butter in the microwave with the mug for awhile.
2. 2Step 2: Warm the milk to a little above room temperature but no hotter than 110 degrees. Stir in the 1/3 cup flour, tablespoon sugar, and yeast and let it sit to turn bubbly.
3. 3Step 3: Cream the butter and sugar until soft and smooth. I use a stand mixer. Add egg yolks one at a time and beat well.
4. 4Step 4: I make this like cake: add some flour, then the yeast sponge, then some more flour, then some of the milk, more flour, add vanilla, rum, and lemon zest, followed by the last of the flour. Let continue to mix until it pulls away from the side of the bowl. If it continues to stick to bowl, add more flour 1 tablespoon at a time.
5. 5Step 5: Once it begins to form a ball, I switch to the dough hook but you could also hand-knead. If doing the latter, I would do it with oiled hands to avoid incorporating much more flour. Knead until smooth and put in a covered bowl to rise until doubled in bulk.
6. 6Step 6: Once the dough has risen, punch it down and separate into golf-ball-sized portions. Roll each until smooth. If you are filling them now, press onto board and flatten out, stretching edges slightly. Put about 1-2 tablespoons plum butter in the center, and pinch the sides up to purse the jam. Seal the pinched dough closed and lay seam side down. If you are not doing this, you can flatten the ball slightly and set to rise.
7. 7Step 7: Let the doughnuts rise again until nearly doubled in bulk.
8. 8Step 8: Fry in hot vegetable oil until dark brown. If you haven't filled them before frying, make a hole inside the doughnut with a chopstick and pipe the jam inside.
9. 9Step 9: Dust or roll in powdered sugar and serve as soon as they are cool enough to eat.