Using the bowl of a stand mixer fitted with the dough hook, add flour, sugar, salt and yeast; stir until well combined.
Using a separate bowl, whisk together the egg, milk, melted butter and vanilla extract.
Pour liquid ingredients into bowl with dry ingredients; mix on low speed for 1 minute. Let dough rest for 5 minutes.
Increase speed to medium and mix for 4-5 minutes.
Spray a large bowl with nonstick cooking spray. Transfer dough into oiled bowl, turning dough over to coat. Cover and let dough rise in a warm place for 1½ hours, or until approximately doubled in size.
Working on a lightly floured surface, roll dough out to ¼” thickness.
Using a 3” round cookie cutter, cut out pieces of dough. Use a 1” round cutter to cut out the center of each donut. (Tip: Save the centers to make donut holes!) Spray wax paper or parchment paper with nonstick cooking spray. Place donuts on top.
Spray tops of donuts with cooking spray and cover lightly with plastic wrap. Let donuts rise 40-50 minutes, or until approximately doubled in size.
Meanwhile, pour the oil into a large deep skillet or Dutch oven. Place over medium heat and bring oil to 350°F.
Working in batches of 2-3 at a time, gently place donuts into oil and fry ~1 minute per side, or until golden brown. Remove donuts from oil and place on a paper-towel lined plate to cool.
For the Glaze
Using a wide, shallow bowl, add confectioner’s sugar, milk and pink food coloring gel; stir until well combined. (Note: If glaze is too thick, add ~1 tsp of additional milk. If glaze is too thin, add ~¼ cup of additional confectioner’s sugar.)
Working with 1 donut at a time, dip the donut into the glaze and then add sprinkles on top.