Combine yeast and 3 tablespoons warm water in a bowl and let stand until foamy, 5 to 10 minutes.
In the bowl of an electric mixer fitted with he dough-hook attachment, combine flour, sugar, dry milk, and salt on low speed.
Add the yeast mixture, eggs, rum, and zest to the bowl. Mix on medium speed. Add up to 1/4 cup warm water until the dough pulls away from the sides of the bowl. Knead for 10 minutes on medium speed.
While mixing, add butter 1 tablespoon at a time; continue adding the rest of the butter until well incorporated.
Remove the dough to a lightly floured surface and knead into a ball. Place in a lightly oiled bowl, cover with plastic wrap, and leave in a warm place until doubled in size, about 1 hour.
With your fist, knock down the dough and remove to a lightly floured surface. Shape the dough into a rough rectangle, and place on a towel-lined baking sheet. Refrigerate until chilled, about 30 minutes.
On a lightly floured surface, roll the dough into a 1/4-inch-thick disk. With a 3 1/2-inch round cookie cutter, cut the doughnuts. Using a spatula, return the doughnuts to towel-lined baking sheet. Roll the scraps into a ball and repeat the cutting process. Place the baking sheet in a warm place, and let dough rise until doubled in size, about 1 hour.
Heat vegetable oil in a Dutch oven or large stockpot until it reaches 325 degrees F. Fry the doughnuts, several at a time, until deep golden brown, 2 to 3 minutes per side.
Remove doughnuts from oil with a slotted spoon, and roll immediately in granulated sugar. Transfer doughnuts to paper-towel-lined baking sheet. Using a knife steel, poke a hole in the side of each doughnut. Fill a pastry bag fitted with a medium round tip, and fill doughnuts with jam or creme patissiere. Serve warm, or store in airtight container at room temperature for up to 1 day.
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