In a large bowl, beat butter and powdered sugar until smooth and creamy.
Add vanilla extract, then mix in the flour and salt until just combined.
Shape the dough into a ball, wrap it in plastic wrap, and refrigerate for 30 minutes.
Preheat the oven to 350°F (175°C). Roll out the dough on a lightly floured surface to about 1/4-inch thickness.
Use cookie cutters to cut shapes, then place on a parchment-lined baking sheet.
Bake for 12–15 minutes, until the edges are lightly golden. Let the cookies cool on a wire rack.
In a medium bowl, whisk together powdered sugar, milk, and extract until smooth.
Adjust milk as needed to reach a spreadable consistency, and add food coloring if desired.
Spread or pipe frosting over each cooled cookie, and add sprinkles if desired. Let the frosting set before serving.