Preheat your oven to 350 degrees F. Line a baking sheet with parchment paper or a silicone mat. Set aside.
In a medium mixing bowl, stir together the flour, baking powder, and salt. Set aside.
In a large mixing bowl (or in your stand mixer), beat together butter and sugar with a hand mixer until fluffy, about 1 minute. Add the egg and vanilla and mix until combined, scraping the sides of the bowl as needed. Add the dry ingredients to the wet ingredients and mix until just combined.
Use a small cookie scoop (or 1 tablespoon measuring spoon) to portion the dough onto the prepared baking sheet. Use your hands to roll the dough into balls. Press your thumb into the center of each ball to create a well. (It's okay if the edges crack a little.)
Add 1/2 teaspoon of raspberry preserves to the well in each cookie. You may need a little more or less depending on how big the thumbprints are.
Bake for 13 to 16 minutes, or until the cookies are set and just beginning to turn golden around the edges. Remove from the oven and cool on the baking sheets for 5 minutes. Then transfer to a wire rack to cool completely.
Glaze
In a small mixing bowl, use a spoon to mix the powdered sugar, milk, and vanilla until smooth. The glaze should easily drizzle off the spoon without being too liquidy. (If needed, add more powdered sugar to thicken, or more milk to thin the glaze.)
Using the spoon, drizzle a little of the glaze over the cooled cookies in a criss-cross pattern. Let the glaze dry for 30 minutes to set.
Notes
Store any leftovers in an airtight container for up to 5 days. You can also freeze these cookies before or after baking.
To freeze the dough: Form the dough into a square, wrap it in parchment paper or plastic wrap, and chill in the fridge for 1 hour. Then wrap it in plastic wrap (yep another layer) and put it in a freezer storage bag. Freeze for up to 3 months. Thaw in the fridge overnight or on the counter for 1 hour until soft enough to slice. Cut into 32 pieces, roll into balls, and bake as directed.
To freeze baked cookies: Let the cookies cool completely. If adding glaze, do so and then let the glaze set up all the way. Place the cookies in a freeze storage bag and freeze for up to 3 months. (I like to put waxed or parchment paper between the layers of cookies.)