Step 1

Mix: In a medium bowl, whisk the flour, sugar, yeast, and salt. Add the milk, oil, egg, vanilla, and zest if using, and mix with a wooden spoon until a rough, wet dough forms.

Step 2

Knead: Place the dough on a well-floured surface, dust the top of the dough with flour, and knead for about 2 minutes. If it remains sticky while kneading, add more flour 1 tablespoon at a time. It should be tacky but not too sticky to knead.

Step 3

Rise: Oil the bowl you just used and place the dough back inside. Cover with a towel and let it rise until doubled in size, about 1 hour.

Step 4

Roll dough: Line a baking sheet with parchment paper. On a floured surface, roll out the dough to 1/2 inch thick.

Step 5

Cut donuts: Using a 2 1/2-inch round cookie cutter or the rim of a glass, cut out as many rounds as you can and place them on the baking sheet. Re-roll the remaining scraps and repeat. You should have 14 to 16 rounds total.

Step 6

Second rise: Cover the baking sheet and let the donuts rise until they puff up slightly, about 30 minutes, depending on the temperature of the kitchen.

Step 7

Prep for baking: While the donuts rise, preheat the oven to 375°F.

Step 8

Bake: After the donuts have risen, bake them for 10 minutes, until golden.

Step 9

While the donuts bake, melt the butter. When the donuts come out of the oven, brush each of the donuts with the melted butter. Proceed to COAT step.

Step 10

Prep for frying: Add vegetable oil to a medium pot, at least 2 inches deep. Heat the oil to 350°F. Line a baking sheet with paper towels to drain excess oil.

Step 11

Deep-fry: Place a few donuts at a time in the hot oil, avoiding overcrowding. Fry for a minute on one side; then flip and fry on the second side for another minute, until golden brown. This is a fast process, so watch carefully. Once fried, transfer the donuts to the prepared baking sheet. Proceed to COAT step.

Step 12

Coat donuts: Place the sugar in a gallon-sized resealable plastic bag. Working one at a time, place each donut in the bag of sugar, seal shut,and shake it around to coat the donut. Return the donut to the baking sheet. Repeat with all donuts.

Step 13

Fill donuts: Fit a piping bag (or a plastic bag) with a decorating tip with a 1/4-inch-wide opening or use a plastic squeeze bottle. Fill the bag or bottle with the jam of your choice. Using a paring knife, cut a 1-inch horizontal slit in the side of each donut, slicing through to the center. Place the decorating tip in the hole and squeeze until the jam starts to ooze out of the donut (some spillage is okay). Repeat with the remaining donuts. The donuts taste best if served immediately, but they will maintain their freshness for several hours.