In a medium saucepan, cook the butter over medium heat, stirring occasionally, until blondish brown, 5 to 8 minutes. Transfer to a mixing bowl and let cool to room temperature.
In a separate mixing bowl, whisk together the flour, baking soda, baking powder and salt. Add the white chocolate and freeze-dried raspberries. Use a flexible spatula, stir to evenly distribute ingredients.
To the cooled butter, add both sugars, the egg and egg white and vanilla and whisk until very smooth and pale brown, at least 1 minute.
Add the dry ingredients to the wet ingredients, and stir just until combined. Scoop the dough into 8 to 10 even portions (90 to 125 grams each). You also can form smaller balls. Place the dough balls onto a large plate or quarter sheet pan. Cover with plastic wrap and refrigerate overnight and up to 24 hours.
When ready to bake, bring the cookie dough to room temperature. Line 2 to 3 baking sheets with parchment paper. Place the dough balls 3 inches apart (3 to 4 per sheet) and flatten each to an inch thickness with the palm of your hand.
Heat oven to 350 degrees, with a rack in the center.
Bake one pan at a time, rotating halfway through, until the cookies have browned slightly at the edges and look dry in the centers, 12 to 15 minutes. While they’re still hot, press a shard of white chocolate in the center of each cookie, then sprinkle with more crumbled freeze-dried raspberries.
Let the cookies cool slightly before transferring to a wire rack to cool completely. The cooled cookies will keep fresh in an airtight container for up to 3 days.