Preheat the oven to 450 degrees. Spray 2 (6-count) doughnut pans with cooking spray. Place a wire cooling rack over a piece of waxed paper.
Chop 1 bag of jelly beans with a sharp, heavy knife or chop them in a small food processor. Spread them out on a cutting board to dry out a little while you prepare and bake the doughnuts.
Whisk together the flour, baking powder, and salt in a medium mixing bowl. Add the milk, sugar, oil, egg, and 1 teaspoon vanilla and whisk until it forms a thick batter. Divide the batter among the doughnut pans, filling about half way. I find it easiest to put the batter into a food storage bag, snip off the tip, and pipe the batter into the doughnut holes. Press 10 jelly beans into the batter of each doughnut, keeping them away from the edges. Bake for 7-9 minutes or until golden brown. Run a knife around the edges of each doughnut and let cool in the pan for 2-3 minutes before transferring to the wire cooling rack. Let cool completely.
Put the powdered sugar in a mixing bowl with the vanilla and melted butter. Add milk, 1 tablespoon at a time, and stir until it reaches a thick but pourable consistency. Add a drop or two of the food coloring. Drizzle the frosting over the doughnuts and sprinkle on the chopped jelly beans. Let the frosting set for about 10 minutes before serving.