Line 2 baking sheets with parchment paper. Melt the butter in a small skillet or saucepan over medium heat. Cook, swirling the pan, until brown flecks appear and the butter smells nutty, about 7 minutes. Pour into a heatproof bowl. Freeze, stirring occasionally, until the butter is spreadable, 35 to 40 minutes. Meanwhile, whisk the flour, baking powder, cardamom and salt in a medium bowl; set aside.
Beat the browned butter and 1/2 cup sugar in a large bowl with a mixer on medium-high speed until light and fluffy, about 3 minutes. Beat in the egg until combined. Reduce the mixer speed to low and beat in the flour mixture until the dough comes together.
Roll the dough into 1-inch balls, then roll in the remaining 1/4 cup sugar. Arrange about 1 1/2 inches apart on the prepared baking sheets. Use the handle of a wooden spoon to indent the centers, pinching together any large cracks (a few small cracks are fine). Refrigerate until firm, 20 to 30 minutes.
Position racks in the upper and lower thirds of the oven; preheat to 375 degrees F. Fill each indentation with 1/2 to 1 teaspoon jam. Bake, switching the pans halfway through, until the cookies are set and lightly browned on the bottom, 20-25 minutes. Let cool completely on the baking sheets.