Preheat oven to 350°F (176° C) and line a baking sheet with parchment paper or a silicone baking mat.
Mix wet ingredients: In a large mixing bowl, use an electric mixer to cream the butter and sugar until light and fluffy, about 2-3 minutes. Add the egg, vanilla extract and sourdough starter discard and beat until combined.
Add dry ingredients: Add the flour, baking powder and salt to the wet ingredients and use the electric mixer to incorporate them until just mixed. Use a spatula to scrape the sides of the bowl, making sure all of the ingredients have been combined.
Shape the cookies: Place ⅓ cup of granulated sugar in a small bowl. Use a cookie scoop to measure out a 1 tablespoon size portion of cookie dough and shape into a ball. Roll it in the granulated sugar to coat. Place the dough ball on the baking sheet and use your thumb or the back of a measuring spoon to press an indentation into the dough ball. Spoon ½ teaspoon of jam into the indentation. Repeat with remaining dough.
Bake: Bake the cookies for 13-15 minutes. Remove them from the oven and allow them to cool on the baking sheet for 10 minutes before transferring to a cooling rack. Once cookies have cooled completely, dust the tops with powdered sugar.