Using an electric mixer, beat butter and sugar at high speed until light and fluffy, about 2 minutes.
Add peanut butter and beat until thoroughly incorporated, about 30 seconds. Scrape down bowl, add the egg, and beat until smooth, about 20 seconds.
In a large bowl, whisk together flour, salt, and baking powder.
Add dry ingredients to butter-sugar mixture and beat at low speed until just combined, about 1 minute. Scrape down bowl, then add milk and beat at low speed until just combined, about 30 seconds.
Gather and press dough together, then divide into 2 equal pieces. Using a sheet of waxed paper as an aid, form each piece of dough into a log about 1 1/2 inches in diameter. Wrap logs tightly in plastic wrap and refrigerate until firm, at least 2 hours and up to 2 days. (Dough can be made ahead and frozen in an airtight bag for up to 6 months).
Preheat oven to 400°F and lightly butter 2 large baking sheets.
Cut each log crosswise into 1/8-inch-thick slices and arrange slices 1/2 inch apart on prepared baking sheets. Top each cookie with a 1/2-teaspoon dollop of jam.
Bake cookies until edges are lightly browned and centers are set, about 10 minutes. Cool on sheets 2 minutes, then transfer to a rack to cool completely. (Cookies will keep, stored in a covered container at room temperature, up to 2 days.)
