01 - Combine warm milk and yeast in a small bowl. Let sit for 5 minutes until the mixture becomes foamy, indicating the yeast is active.
02 - In a large mixing bowl, whisk together the flour, sugar, and salt until evenly combined.
03 - Add melted butter, eggs, vanilla extract, and the activated yeast mixture to the dry ingredients. Mix until a shaggy dough forms.
04 - Knead the dough by hand or with a stand mixer fitted with a dough hook for 6 to 8 minutes, until smooth and elastic.
05 - Place the dough in a greased bowl, cover with plastic wrap or a towel, and let rise in a warm place for 1 hour or until doubled in size.
06 - Divide the 1/3 cup of granulated sugar into three separate bowls. Add a few drops of purple, green, and yellow food coloring to each bowl. Rub with your fingers until the sugar is evenly colored. Set aside.
07 - Turn the risen dough onto a floured surface. Roll out to 1/2-inch thickness. Cut out 3-inch rounds using a donut cutter, or use a glass for the outer circle and a small cutter for the centers.
08 - Place the cut donuts on parchment-lined baking trays. Cover lightly with a towel and let rise for 30 minutes until puffy.
09 - Heat vegetable oil in a large, heavy-bottomed pot to 350°F.
10 - Fry donuts in batches for 1 to 2 minutes per side, until golden brown. Remove with a slotted spoon and drain on paper towels.
11 - Whisk together powdered sugar, milk, and vanilla extract until smooth and runny.
12 - Dip the warm donuts into the icing, allowing excess to drip off. Immediately sprinkle with the colored purple, green, and yellow sugars in sections or stripes to create a traditional Mardi Gras pattern.
13 - Let the iced donuts set for 10 to 15 minutes before serving.