Place butter in a microwave safe bowl, and melt, but not entirely, you should still see a few chunks.
Add almost melted butter to a large bowl with the shortening. Add white sugar and brown sugar and beat on medium-high speed until well combined. Stop once to scrape the sides and bottom of the bowl with a spatula.
With the mixer on low-medium speed, beat in the egg, egg yolk, lemon zest, and lemon juice.
In a separate bowl combine dry ingredients: flour, salt, and baking powder, whisk until just combined.
Add dry ingredients to the mixer and beat on low just until combined. Do not over-mix.
Using a spatula, gently stir in the white chocolate chips and freezer dried raspberries.
Cover the bowl tightly and chill for 2 to 8 hours.
Preheat oven to 325F.
Line a pan with parchment paper.
Scoop 1½ tablespoon of cookie dough, roll into a ball and slightly flatten it with your hand. Place cookie on the prepared sheet, 6-8 cookies per pan so they have plenty of room.
Bake for 10-12 minutes, making sure you don't over bake, better to under bake for a cookie-dough like center. Look for a golden hue on the sides of the cookies.
Optional: after removing the cookies from the oven, press a few chocolate chips onto the top.
Cool cookies for 5-7 minutes on the baking sheet after that transfer to a cooking rack.