Preheat 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 flour, baking powder, and salt. Set aside.
In a large mixing bowl (or in your stand mixer), beat together butter and sugar until fluffy, about 1 minute. Add egg and vanilla; 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 your prepared baking sheets. 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 about 1/2 teaspoon 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. Transfer to a wire rack to cool completely.
In a small mixing bowl, use a spoon to mix together powdered sugar, milk, and vanilla until smooth. The glaze should easily drizzle off the spoon with being too liquidy. (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. (You can eat them right away if you want.)