Preheat your oven to 350 degrees and line two baking sheets with parchment paper. In a stand mixer, cream together the softened butter, powdered sugar, and vanilla for two to three minutes until the mixture turns pale and fluffy. This step builds the airy texture that makes these cookies so tender.
Add the salt and mix until just combined, then gradually add the flour about half a cup at a time. Mix well after each addition until the dough comes together and holds its shape when pressed. Don't rush this part, or the flour won't fully incorporate.
Scoop the dough into even tablespoon portions and roll each one into a smooth ball with your hands. Roll every ball in your sprinkles, pressing gently so they stick evenly across the surface. Space the balls about an inch and a half apart on your prepared baking sheets.
Using your thumb, press down gently into the center of each dough ball to create a small indentation. Go slowly here, since the dough is tender and can crack easily along the sides if you press too hard or too fast.
Bake the cookies for 10 minutes, then pull them out and use a melon baller or small round tool to deepen the indentation slightly. Return them to the oven for another 7 to 8 minutes, until the cookies look set but not browned.
Move the cookies to a wire rack and let them cool all the way through before you even think about icing them. Warm cookies will melt the icing right out of their little wells, so patience really matters at this stage.
Whisk together the powdered sugar, corn syrup, vanilla, melted butter, and a tablespoon of milk until smooth, adding a touch more milk if needed. Divide the icing into small bowls and stir in a few drops of gel food coloring to match your sprinkles.
Spoon a small amount of colored icing into the center of each cooled cookie, filling the indentation without overflowing. Let the icing set at room temperature for about an hour before serving or stacking the cookies.