For the starter: Mix the rice flour and milk in a small microwave-safe bowl until smooth. Microwave for 20 seconds and stir (the mixture will be very thick). Microwave for 10 seconds, then stir. The mixture should now be very sticky (it will pull like a rubber band when stretched upwards with a spoon) and have gone from white to translucent. If not, microwave for 10 seconds more. Set aside to cool slightly.
For the doughnuts: Combine the rice flour, granulated sugar, matcha powder, baking powder and salt in the bowl of an electric mixer. Mix with the paddle attachment on low just to combine. Whisk together the milk, butter, vanilla and egg in a spouted measuring cup. Add to the mixer bowl and beat on medium to combine. Scrape in all of the starter and mix until the dough comes together. Mix on high speed until very smooth, 1 to 2 minutes. (The dough will not clean the sides of the bowl and be slightly sticky to the touch.)
Dust your work surface with rice flour and empty the dough onto the counter. Knead until soft and smooth (almost like Play-Doh). Cut 12 (4-by-4-inch) squares of parchment. Roll the dough to a scant 1/2 inch and cut as many circles as you can with a 1 1/4-inch ring cutter. Reroll to cut more circles. Group 6 dough rounds on a piece of parchment in the shape of a circle, with the pieces touching and a hole in the middle. Pat gently to seal the pieces together. Repeat with the remaining circles to get about 12 doughnuts.
Meanwhile, heat about 2 inches of oil in a large Dutch oven to 340 degrees F.
To fry, slide the doughnuts, still on the paper, into the hot oil in batches of 4. Let fry about 30 seconds, then remove the papers and discard. Continue to fry until cooked, 5 to 6 minutes total per batch. Let cool completely before glazing.
For the glaze: Whisk the confectioners' sugar and matcha in a small bowl to combine. Whisk in the lemon juice, vanilla and 3 tablespoons of the milk to make a smooth, thick glaze. (Gradually add 1 tablespoon more milk if needed.) If the glaze is still too thick (it should run leisurely from a spoon), add more milk a teaspoon at a time to reach the desired consistency.
Dip the top of each doughnut in the glaze and then into the sprinkles if using. Place on a baking rack to dry and harden.