Preheat oven to 350°F. Cover two large baking sheets with parchment paper.
Beat the butter and sugar on medium speed in the bowl of a standing mixer. Add the egg and vanilla; beat again. Scrape down the sides of the mixing bowl with a large spatula. Add the flour, baking powder and salt. Mix until just combined. Turn the dough out onto a flour-dusted work surface and knead briefly. Shape dough into a ball.
Divide the dough ball into four equal portions. Roll each portion under your palms to a baton 12-inches in length. Place two batons, well-spaced, on each baking sheet. Use the handle of a wooden spoon to make an indention down the centers of each baton. Fill each depression with jam.
Bake each pan for 15-20 minutes, or until the dough is puffed and fragrant with light brown edges. Cool for 5 minutes then cut each baton diagonally into 1-inch slices. Cool completely on wire racks.
In a small bowl stir together the sugar and milk. Pipe or drizzle the glaze over the cookies. Let stand until set, about 1 hour.
Store in airtight containers or gift in a large festive holiday tin.