In a large bowl, or stand mixer bowl, add warm water, and sprinkle yeast on top. Let sit for 5 minutes.
Heat the milk and butter to about 110 degrees, or until butter is melted. Add to yeast mixture. Add in the sugar, salt, and eggs. Mix until combined.
Add in half the flour, and combine using a dough hook. Add remaining flour. Knead with dough hook for about 5 minutes, until the dough comes clean from the sides of the bowl. If needed, add more flour, 1/2 cup at a time.
Place in a greased bowl, cover, and let rise for 1-2 hours, or until doubled in size.
Dump the dough out onto a floured surface. Roll into 1/2 inch thickness. Cut the donuts out with a cookie cutter or large glass. Cut a slit in the center and form a hole. Place on a parchment lined baking sheet. Cover, and let rise an additional 30 minutes.
Heat about 2 inches of oil in a deep skillet or pot to 350 degrees. Fry donuts, 2 or 3 at a time, for about 2-3 minutes per side. Transfer to a wire rack, and cool for 20 minutes.
Meanwhile, make the glaze by combining the confectioners sugar, extracts, food dye, and milk. Start with 2 TBSP. You want the glaze to be thick enough to coat the donut. Add sprinkles on top.
