Preheat oven to 350° F. Line two cookie sheets with parchment paper or a Silpat. (If you don't have parchment or a Silpat, you can skip this step. It just makes the cookies easier to remove from the cookie sheet.)
Whisk together all-purpose flour, baking powder and salt in a small mixing bowl and set aside.
In the bowl of a stand mixer or a large mixing bowl with a hand mixer, beat softened butter on medium speed until creamy, about two minutes.
Zest and juice the lemons.
Add the lemon zest to the sugar. Rub the zest into the sugar with your fingers until it's combined. (Doing this method releases the oils in the zest so they are absorbed into the sugar.)
Add the lemon zest with sugar and lemon juice to the butter and beat on medium speed until light and fluffy. (Save the rest of the lemon juice for the glaze.)
Add egg to the bowl and beat until combined. Scrape down the bowl with a rubber spatula as needed.
Add the flour mixture to the bowl and beat on low until just combined.
Add the raspberries to the bowl and carefully fold into the cookie batter. (Don't worry if the raspberries break while folding them in.)
With a medium cookie scooper, (or 1 1/2 tablespoons), scoop cookies onto cookie sheet two inches apart. (Cookie dough does not need to be chilled.)
Bake cookies for 10-12 minutes until the cookies are just set and are starting to turn golden brown on the edges. (If you're using a smaller cookie scoop and have smaller cookies, bake for about 8-10 minutes.)
Transfer cookie sheet to a cooling grid and cool for 5 minutes.
Remove cookies from cookie sheet and transfer to cooling grid to cool. Make sure the cookies are completely cool before glazing.