If there are any large pieces of fruit in the preserves, cut them into smaller pieces.
Using an electric mixer on medium speed, beat the preserves, butter, and vanilla until thoroughly combined.
Transfer the butter mixture to a sheet of plastic wrap. Shape into a thin disc or a log. Wrap and freeze for 1 to 2 hours, or until firm.
Preheat oven to 325°F (160°C). Line baking sheets with parchment paper or silicone liners.
Whisk together the flour, sugar, and salt. Add the butter mixture, and mix with a pastry blender or a fork until the mixture resembles fine crumbs.
Use your hands to continue mixing the dough until it forms a cohesive dough.
Divide the dough in half. Place one half on a lightly floured surface. Roll to a 1/4-inch thickness. Use a 2-inch round cookie cutter to cut out cookies**. Repeat with the other half of the dough.
Place the cookies on the prepared pans, leaving about an inch between cookies. Bake 14 to 16 minutes, or until the edges of the cookies begin to brown. Cool the cookies on the pan for 5 minutes. Then transfer the cookies to a wire rack to cool completely.
To make the glaze:
Place the preserves in a microwave-safe bowl. Heat at half power for 30 seconds, or until the preserves are thinner and stirrable. If there are any large pieces of fruit in the preserves, cut them into smaller pieces.
Whisk together the warm preserves, confectioners' sugar, and 1 tablespoon milk. Add more milk, 1 teaspoon at a time, until the glaze is smooth and spreadable.
Spread the glaze on top of the cooled cookies. Sprinkle with sanding sugar. Allow the glaze to set before serving.
Notes
*Be sure you measure your flour accurately so that the dough will come together well. If you don't use a digital scale, then spoon the flour into the measuring cups instead of scooping down into the flour container. That will help prevent using too much flour. See my tips for How to Measure Flour if you need a refresher.
**Feel free to make these smaller or larger if you like. Smaller cookies will bake more quickly, while larger ones will require a bit more baking time.
To store: Once cooled, place the cookies in an airtight container. While they'll be at their best within a day, they should keep up to 3 days. For longer storage or if you have a particularly warm kitchen, place the container in the refrigerator. Just be sure it's airtight so the cookies don't become dry.
To freeze: These cookies freeze best without the glaze. If you want to freeze them with the glaze, first place them on a rimmed baking sheet in the freezer to allow the glaze to freeze thoroughly. Then place the cookies in a single layer or between layers of waxed paper in an airtight, freezer-safe container. Properly stored, the cookies should keep up to 3 months. Thaw them overnight in the refrigerator or for an hour or two at room temperature.