Line the bottom of an 8-inch baking pan with parchment paper; spray the paper lightly with cooking spray.

Mix together flour, baking powder, and 1 pinch salt in a bowl.

In the bowl of a stand mixer fitted with the paddle attachment, beat 1/2 cup butter and white sugar together until light and fluffy. Add egg and 1 teaspoon vanilla extract and beat until well combined.

Gradually add flour mixture. Fold in white chocolate chips. Press dough into the prepared baking pan; refrigerate for 2 hours.

Preheat the oven to 350 degrees F (175 degrees C). Line a baking sheet with parchment paper; set aside.

Bake cookie bars in the preheated oven until lightly browned around the edges and set in the center, about 15 minutes. Set the pan on a wire rack to cool completely, about 20 minutes.

Meanwhile, toss berries with vegetable oil in a bowl. Arrange berries in a single layer on the prepared baking sheet.

Roast berries until very soft, turning halfway through baking time, about 25 minutes. Transfer berries to a small mixing bowl to cool completely before proceeding, about 20 minutes. Discard any berry bits that are burned. Once cooled, chop them up a bit.

Place 1 cup butter in the bowl of a stand mixer fitted with the paddle attachment and beat until very smooth. Add berries; beat on medium speed until berries are broken up and mixture is well combined.

Turn mixer to low and gradually add 2 1/2 cups confectioners’ sugar, beating until well combined. Add 1/2 teaspoon vanilla extract and 1 pinch salt, then beat on high speed until frosting is very fluffy, 3 to 4 minutes.