First, make the donuts. Position a rack in the center of the oven and preheat the oven to 350°F.
Mix the dry ingredients. In a small bowl, whisk together the mochiko sweet rice flour, baking powder, and kosher salt.
Mix the wet ingredients. In a medium bowl, whisk together the butter, sugar, and egg yolks until combined. Add the ube jam, ube extract, and purple food coloring and whisk until combined. Gradually whisk in the milk until just combined.
Add the dry ingredients. Gradually mix in the dry ingredients into the wet ingredients until just combined. Pour the batter into a liquid measuring cup or pitcher with a spout. Prep the donut pan. Lightly spray the cavities of a donut pan with cooking spray. Use the liquid measuring cup to pour the batter into the cavities, filing each cavity completely up to its brim with batter. Bake the donuts. Bake for 35 minutes. When done, the top of each donut should bounce back when gently pressed and a skewer inserted into the center of the cake should come out with a few crumbs attached. Cool the donuts in the pan for 5 minutes, then turn out to cool completely on a wire rack before glazing. Make the glaze. In a medium bowl, whisk together the ube jam and milk. Add the confectioners' sugar, ¼ cup (1 ounce or 28 grams) at a time, whisking until completely incorporated and smooth before adding the next batch of sugar. Once the glaze is smooth, whisk in the ube extract. The glaze will be thick at this point—loosen it by whisking in the remaining ½ teaspoon milk.
Glaze the donuts. Working quickly, dip the top of a donut in the glaze and quickly flip it up so that the glazed side is upright. Place on a wire rack to let any excess glaze to drip off. Garnish immediately with sprinkles. Repeat with remaining donuts.
Serve and store. Serve at room temperature. You can serve immediately after glazing, or wait 1 to 2 hours for the glaze to dry and harden. The donuts are best on the day that they're made, but can be stored in an airtight container at room temperature for up to 2 days.