01 - Combine lukewarm milk and yeast in a small bowl. Let sit for 5 minutes until mixture becomes foamy, indicating the yeast is active.
02 - Whisk flour, sugar, and salt in a large mixing bowl. Add eggs, softened butter, vanilla extract, and the activated yeast mixture. Mix until a sticky dough forms.
03 - Turn dough onto a lightly floured surface. Knead for 8 to 10 minutes until smooth and elastic, adding minimal flour only if necessary to prevent excessive sticking.
04 - Place dough in a greased bowl, cover with plastic wrap or a clean towel. Let rise in a warm, draft-free area until doubled in size, approximately 1 hour.
05 - Punch down risen dough to release air. Roll out to 1/2-inch thickness. Use a donut cutter or two round cutters (3-inch and 1-inch diameter) to cut out donuts and holes.
06 - Arrange cut donuts on parchment-lined baking sheets, leaving space between each. Cover and let rise for 30 minutes until puffy.
07 - Add a few drops of food coloring to each bowl of sugar (purple, green, and yellow). Rub with fingers until evenly colored and vibrant. Set aside.
08 - Heat vegetable oil in a large pot to 350°F. Fry donuts in batches for 1 to 2 minutes per side until golden brown. Remove with a slotted spoon and drain on paper towels.
09 - Brush warm donuts lightly with melted butter. Dip or sprinkle with colored sugar in alternating bands to create the classic Mardi Gras purple, green, and gold pattern.