Preheat the oven to 350°F and line a baking sheet with parchment paper.
Beat the softened butter and powdered sugar together until smooth and creamy.
Mix in the vanilla extract until fully incorporated.
Add the flour and salt and mix just until a soft dough forms.
Scoop the cookie dough and roll it into smooth, evenly sized balls.
Lightly dust the inside of the wooden cookie stamp with flour, using a pastry brush to get flour into all the details.
Lightly dust the outside of each dough ball with flour as well.
Place one dough ball into the center of the floured cookie stamp.
Press the dough firmly into the stamp, pushing it into the design and smoothing the surface so it is level with the edges of the stamp.
Gently loosen the dough from the sides of the stamp with your fingers.
Turn the stamp upside down over the prepared baking sheet and tap firmly on the back until the stamped cookie releases.
Repeat with the remaining dough, re-flouring the stamp as needed to prevent sticking.
Chill the stamped cookies in the refrigerator for about 10 minutes.
Bake for 6 to 7 minutes until the cookies look set but are still pale.
Remove the baking sheet from the oven and spoon about ½ teaspoon of jam into the center of each cookie.
Return the cookies to the oven and bake for another 6 to 7 minutes, until the edges are lightly golden.
Let the cookies cool on the baking sheet for a few minutes, then transfer to a wire rack to cool completely.