Chop raspberries into rough 1/4-inch pieces and place in a single layer on a paper towel-lined plate. Pat gently to remove excess moisture. Freeze for 10 minutes while you make the dough. This prevents them from bleeding into the batter.

Beat butter and sugar on medium speed until light and fluffy, about 3 minutes. Scrape down the bowl. Add egg, lemon zest, lemon juice, and vanilla. Beat until combined, about 1 minute.

Whisk together flour, baking powder, and salt. Add to butter mixture on low speed just until no dry streaks remain. Do not overmix.

Remove raspberries from freezer. Fold in three-quarters of them gently with a spatula, leaving streaks of pink is fine. Reserve remaining raspberries for pressing on top.

Scoop 1.5 tablespoon mounds of dough onto parchment-lined baking sheets, spacing 2 inches apart. Press a few reserved raspberry pieces onto each cookie. Chill in refrigerator 15 minutes while oven preheats to 350°F.

Bake until edges are set and just beginning to turn golden, 12 to 14 minutes. Centers will look slightly underdone. Cool on baking sheet 5 minutes, then transfer to wire rack to cool completely.

Whisk powdered sugar and lemon juice until smooth and pourable, adding more juice by the teaspoon if needed. Drizzle over cooled cookies. Let set 15 minutes before serving.