Preheat the oven to 350°F. Spray a 9-inch high-sided pan or springform pan with nonstick cooking spray. If using a springform pan, place a piece of foil underneath or set the pan on a sheet tray to catch any potential drips.
In a medium bowl, whisk together the melted butter and brown sugar. Spread this mixture evenly into the bottom of the prepared pan.
Arrange the raspberries in an even layer over the brown sugar mixture, using enough to fully cover the bottom of the pan. Set aside.
In a large bowl, whisk together the flour, baking powder, and salt. Set aside.
In another bowl, beat together the softened butter, granulated sugar, egg, and vanilla extract until fully combined.
Gradually add the flour mixture to the butter mixture and beat until combined. The batter will appear crumbly at this stage, which is normal.
Add the milk to the batter and beat until smooth and fully incorporated.
Spoon the batter over the raspberries, spreading gently and evenly with an offset spatula as the batter will be thick.
Bake for 35 to 40 minutes or until golden brown and a toothpick inserted into the center comes out clean. Oven times may vary, so check for doneness.
Remove the cake from the oven and allow it to cool in the pan for 10 minutes. Run a knife along the edges of the pan to loosen the cake before inverting it onto a serving platter.
Allow the cake to cool completely. Dust with powdered sugar if desired before serving.