Preheat your oven to 375°F (190°C). Spray a 6-cavity donut pan generously with nonstick cooking spray, making sure to coat the center posts thoroughly.
In a large mixing bowl, whisk together the flour, granulated sugar, baking powder, and salt until no lumps remain and the mixture is evenly combined.
In a separate medium bowl, whisk together the milk, egg, melted butter, and vanilla extract until smooth and fully incorporated.
Pour the wet ingredients into the dry ingredients and stir gently with a rubber spatula just until combined. The batter should be slightly thick but pourable, with no streaks of flour visible.
Fold in the rainbow sprinkles using 3-4 gentle strokes to distribute them evenly without overmixing, which can toughen the texture.
Transfer the batter to a piping bag or large zip-top bag with one corner snipped off. Pipe the batter into each donut cavity, filling them about two-thirds full to allow room for rising.
Bake for 9-11 minutes, until the tops spring back when lightly touched and a toothpick inserted into the thickest part comes out clean. The edges should be just beginning to turn golden.
Let the donuts cool in the pan for 5 minutes, then carefully turn them out onto a wire rack set over parchment paper.
While donuts cool slightly, whisk together the powdered sugar, 3 tablespoons milk, and vanilla extract in a shallow bowl until the glaze is smooth and flows easily off a spoon.
Dip the top of each warm donut into the glaze, allowing excess to drip back into the bowl. Immediately sprinkle with additional rainbow sprinkles while the glaze is wet. Let the glaze set for 15 minutes before serving.