Preheat oven to 350 degrees, with racks in upper and lower thirds. In a medium bowl, whisk together flour, baking soda, baking powder, and salt. In a large bowl, using an electric mixer, beat almond butter and butter on medium until smooth. Add brown sugar and beat until fluffy, about 3 minutes. Add egg and vanilla and beat until combined. With mixer on low, gradually add flour mixture and beat just until combined (dough will be slightly crumbly).
Place granulated sugar in a small bowl. Roll level tablespoonfuls of dough into balls and gently roll in sugar to coat. Place on two parchment-lined rimmed baking sheets, 1 1/2 inches apart. Bake until cookies are puffed, 8 minutes. Remove sheets from oven and, with the bottom of a round 1/2 teaspoon measuring spoon, make an indentation in center of each cookie. Bake until set, 5 to 7 minutes. Let cookies cool on wire racks, 20 minutes. Stir jam and spoon 1/2 teaspoon into each indentation.
Store in single layers in air-tight containers, up to 3 days.