Preheat your oven to 350°F (175°C) and line a baking sheet with parchment paper.
In a mixing bowl, cream together the softened butter, granulated sugar, and powdered sugar until light and fluffy using an electric mixer or whisk.
Add the egg and vanilla extract to the butter mixture and mix until well combined.
In a separate bowl, whisk together the flour, baking powder, and salt. Gradually add this dry mixture to the wet ingredients, mixing until just combined.
Gently fold in the fresh raspberries, being careful not to break them apart.
In another small bowl, combine the rolled oats, brown sugar, and a pinch of salt to create the crumble topping.
Use a tablespoon or scoop to portion and drop dollops of the cookie dough onto the prepared baking sheet, leaving space between each.
Sprinkle the crumble topping evenly over the cookies.
Bake in the preheated oven for 12-15 minutes, or until the edges are lightly golden.
Remove from the oven and let cool on the baking sheet for a few minutes, then transfer to a cooling rack to cool completely.