Heat milk in a small pan until warm. Transfer to a small bowl, and add yeast and 2 teaspoons sugar; let stand until foamy, about 10 minutes. Whisk egg, vanilla, and the water in a separate bowl.
Combine flour, salt, and remaining sugar in a mixer bowl. On low speed, add yeast mixture, egg mixture, and melted butter, mixing until just combined. Switch to a dough hook; knead on medium speed until dough forms a ball and pulls away from the sides of the bowl, 18 to 20 minutes. Place in a large greased bowl; cover tightly with plastic. Refrigerate overnight.
Line a baking sheet with parchment; spray with cooking spray. Roll dough out to 1/4 inch thick on a lightly floured surface. Cut out with 1- and 1/2-inch round cutters; place dough on prepared baking sheet, spaced about 1 inch apart. Spray a second sheet of parchment, and lay over dough, sprayed side down. Place in a warm spot and let rise about 1 hour.
Fill heavy-bottomed pot or deep-fryer with 10 cups vegetable oil. Heat to 375 degrees on a deep-fry thermometer. Line a baking sheet with paper towel.
Carefully drop dough into hot oil in batches. Fry, moving doughnut holes around as they cook, until golden brown, 1 to 2 minutes. Transfer doughnut holes to paper towels to drain briefly.
Make glaze: Whisk confectioners' sugar and milk in a large bowl until smooth and honeylike. Spread colored sugars in pie plates. Dip doughnut holes in glaze, place on a wire rack, and let stand about 1 minute, until excess has dripped off but glaze is still wet. Roll in colored sugars; return to rack to set.
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