Preheat oven to 350°F. Line two baking sheets with parchment paper.
Beat butter, granulated sugar, and brown sugar until pale and fluffy, 2–3 minutes.
Mix in beaten egg, vanilla, and almond extract until smooth.
Whisk all-purpose flour, almond flour, baking powder, and salt in a separate bowl.
Stir dry ingredients into the butter mixture until a soft dough forms; do not overmix.
Scoop 1-tablespoon portions, roll into balls, and place 2 inches apart on the sheets.
Press a shallow thumbprint in each. Tuck in a pinch of crumbled almond paste.
Gently press sliced almonds over the tops to adhere and mostly cover the centers.
Bake 12–14 minutes, until edges are light golden and centers look just set.
Warm apricot jam briefly and brush over warm cookies for shine.
Cool completely on racks, then dust with powdered sugar to finish.