These light, delicious doughnuts come to us courtesy of pastry chef Francois Payard, of Payard Patisserie and Bistro in New York City.
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 the dough-hook attachment, combine flour, sugar, dry milk, and salt on low speed.
Add the yeast mixture, eggs, rum, and lemon 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 to add 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 doughnuts. Using a spatula, return 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. Fry 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.