Preheat your oven to 350 degrees F. Line a baking sheet with parchment paper or a silpat. Don't get the raspberries out of the freezer yet, leave them in until the very last minute.
In a large bowl or stand mixer, beat together the butter and shortening for 1-2 minutes, scraping the sides and bottom.
Add the sugar and lemon zest and beat well, another 1-2 minutes, until fluffy.
Add 1 tablespoon lemon juice and the egg. Beat well.
Add 2 cups of flour but don't mix yet. Use a small spoon to stir the baking powder, baking soda, and salt into the flour. Beat until almost combined (It will have formed a dough but there might still be streaks of flour.)
Get the raspberries out of the freezer. Measure out 3/4 cup onto a cutting board and chop roughly with a sharp knife. You want each raspberry to be at least chopped in half but not much more than that. Break up any big clumps.
Carefully stir the raspberries into the dough with a spatula or wooden spoon. Don't over mix; you want there to be sections of lemony dough that don't have raspberries in them. If you mix too much it will turn all the dough pink.
Use a cookie scoop to immediately shape all the dough. You should get about 15 cookies. Add 6-8 to the prepared pan, at least 2 inches apart.
Bake at 350 for 14-15 minutes, or until the edges have barely turned golden. Make sure that the top of the cookie is not too shiny. If it still looks raw in the middle, leave it in for another couple minutes.
Once you have the cookies in the oven, immediately shape the other cookies and place on a plate or another prepared baking sheet, if it will fit in your freezer. Place the plate in the freezer (otherwise the raspberries will start melting and get liquid everywhere. You need them to go in the oven frozen.)
Bake the remaining cookies right away once the oven is free. Bake them the same amount of time.
If you want to save the other cookies for later, transfer the shaped dough to a ziplock freezer bag. You can freeze it for up to 2 months. (I do this with all my cookies! Cookie dough freezes beautifully.) Don't thaw before baking. Just plop them on the pan and throw it in a preheated oven. You might need to increase the baking time by a couple minutes.
Let baked cookies set on the pan for 5 minutes, then transfer to a cooling rack. Let cool completely. You can speed this up by putting them in the fridge or freezer if you want.
Make the glaze: in a small bowl combine 3 tablespoons fresh lemon juice (about 1 lemon) and the powdered sugar. Add more lemon juice or powdered sugar to get the consistency you want. (I like my glaze a little thick.)
Use a spoon to glaze the cookies. Let the frosting set at room temperature. Top each cookie with a fresh raspberry.
These cookies are best served on the day you make them. Store leftover cookies in the fridge.