Preheat oven to 350 degrees F and line 2 large baking sheets with parchment paper.
Mix together the sugar and oil in a large bowl. Add in the milk, flax egg, lemon, and vanilla, and mix until creamy. Stir in the flour, baking powder, and salt to form a soft dough.
Scoop 1 tablespoon portions of dough, roll into balls, and arrange on the prepared cookie sheets about 2 inches apart. Bake for 12-15 minutes, until puffy and firm.
Allow to cool completely, then whisk together the icing ingredients in a bowl (start with 2 tablespoons of milk, and add another if necessary to reach a smooth and dippable consistency). Dip the tops of the cookies in the icing then immediately sprinkle with nonpareils. Allow the icing to set, then enjoy!