Preheat oven to 350 F and line cookie trays with a sheet of parchment paper or silicone baking mat.
In a large bowl using an electric mixer, or in the bowl of a stand mixer fitted with the paddle attachment, cream the butter, sugar until light and fluffy.
Add the egg yolk and vanilla and beat on medium speed until well combined.
Sift the dry ingredients (flour and salt) into the butter mixture. Beat on low speed until large crumbles form.
Add the milk and mix well until a dough forms. The dough should not form a ball, but should be very sticky and have the consistency of a thick buttercream. If you stick your spatula in and pull it out, some batter should stick. If the dough is not sticking, add a little more milk.
Transfer the half of the dough to a large pastry bag fitted with a large star tip.
For the sandwich cookies, pipe the cookie batter into 2-inch long segments, leaving an inch and a half between each cookie. Use a sharp knife to neatly trim the edges.
For the star cookies, hold the piping bag vertically and squeeze until your star shape is about 1-inch in diameter. Use a 1/4 teaspoon or your finger to create a divot in the center of each cookie, so that you can pipe some melted chocolate in after baking.
Repeat with second half of dough.
Bake cookies in the oven until light golden brown on the edges, about 10-15 minutes.
Transfer to a wire rack to cool completely.
While the cookies cool, melt the chocolate in the microwave in a microwave-safe bowl.
To make the sandwich cookies:
Spread a thin layer of jam onto one side of each cookie. Dip one end of the cookie into melted chocolate and lay it on a piece of parchment paper. Add sprinkles over the top for decoration.
To make the star cookies:
Spoon the remaining melted chocolate into a piping bag with a small round tip. Pipe dots of chocolate into the little divot in each star cookie. Add some sprinkl
Notes
These homemade butter cookies keep well in an airtight container for several days.