Preheat the oven to 350°F (175°C) and line a baking sheet with parchment paper.
Cream together the softened butter and granulated sugar in a large bowl until light and fluffy.
Beat in the egg and vanilla extract until well combined.
In a separate bowl, whisk together 2 cups flour, baking powder, and salt. Gradually add this to the butter mixture, mixing until just combined.
Gently fold in the fresh raspberries, being careful not to crush them too much.
For the crumble topping, mix brown sugar, rolled oats, 1/4 cup flour, and cold cubed butter with your fingers until the mixture resembles coarse crumbs.
Drop spoonfuls of dough onto the baking sheet, spacing them about 2 inches apart.
Sprinkle each cookie with a generous amount of crumble topping.
Bake for 15–18 minutes, until the edges are golden and the topping is crisp.
Let cookies cool on the sheet for 5 minutes, then transfer to a wire rack to cool completely.