Whisk confectioners’ sugar, 1/4 cup sprinkles, 3 tablespoons milk, lemon juice, 1 teaspoon vanilla extract, and 1 pinch salt together in a bowl to make a smooth glaze.
Whisk flour, brown sugar, baking powder, and 3/4 teaspoon salt together in a large bowl.
Combine 1 cup milk, egg, and 1 teaspoon vanilla extract in a small bowl. Stir into the flour mixture. Add melted butter and 1/4 cup sprinkles; mix until a soft dough forms.
Heat at least 2 inches of oil in a large pot to 350 degrees F (175 degrees C). Line a baking sheet or wire rack with paper towels.
Scoop a few tablespoons of dough into the hot oil using a small ice cream scoop. Fry donut holes until golden brown, about 1 minute per side. Transfer donut holes to the paper towels using a slotted spoon; let drain and cool slightly. Repeat with remaining dough.
Place a wire rack on a baking sheet. Dip one donut hole at a time into the glaze and place on the wire rack; allow excess glaze to drip off before serving.