01 - Combine lukewarm milk and yeast in a small bowl. Let sit for 5 minutes until foamy.
02 - Whisk flour, sugar, and salt in a large mixing bowl. Add eggs, softened butter, vanilla, and yeast mixture. Mix to form a sticky dough.
03 - Knead dough on a lightly floured surface for 8–10 minutes until smooth and elastic.
04 - Place dough in a greased bowl, cover, and let rise in a warm place until doubled in size (about 1 hour).
05 - Punch down dough. Roll out to 1/2-inch thickness. Using a donut cutter or two round cutters (3 and 1 inch diameter), cut out donuts and holes.
06 - Place donuts on parchment-lined baking sheets. Cover and let rise for 30 minutes.
07 - Add a few drops of food coloring to each bowl of sugar and rub with fingers until evenly colored. Set aside.
08 - Heat oil in a large pot to 350°F. Fry donuts in batches, 1–2 minutes per side, until golden brown. Remove with slotted spoon and drain on paper towels.
09 - Brush warm donuts lightly with melted butter, then dip or sprinkle with colored sugar in alternating bands to mimic Mardi Gras colors.