Preheat oven to 350°F (175°C). Cream the vegan butter, granulated sugar, and brown sugar with an electric mixer on medium speed for about 2 minutes, until light and fluffy.

Add the 3 tbsp applesauce and 2 tsp vanilla, then 1 tbsp almond milk if needed, and mix on low until combined.

Whisk the flour, baking powder, and salt in a separate bowl, then fold into the wet mixture just until no dry streaks remain.

Shape the dough into a disc, wrap in plastic, and refrigerate for at least 30 minutes. Roll the chilled dough to 1/4-inch thickness on a floured surface and cut out shapes.

Place cookies on a parchment-lined baking sheet and bake for 8-10 minutes, until edges are just barely golden. Cool on the sheet for 5 minutes, then transfer to a wire rack.

Whisk together powdered sugar, 1 tbsp almond milk, and 1/4 tsp vanilla. Thin with more milk as needed. Dip cooled cookies, top with sprinkles, and let the icing set for about 30 minutes before serving.