1. Preheat your oven to 350°F and line an 8×8 inch baking pan with parchment paper.
2. In a medium bowl, whisk together the flour, cocoa powder, baking powder, and salt.
3. In a large bowl, mix the melted butter and granulated sugar until well combined. Add the eggs one at a time, followed by the vanilla extract.
4. Gently fold in the dry ingredients into the wet mixture until just combined. Avoid overmixing to keep the brownies fudgy.
5. Fold in the pureed strawberries and chopped strawberries, ensuring they are evenly distributed.
6. Pour the batter into the prepared baking pan and smooth the top with a spatula.
7. Bake in the preheated oven for 25-30 minutes or until a toothpick inserted into the center comes out with a few moist crumbs.
8. While the brownies are baking, prepare the frosting. In a small bowl, beat the cream cheese until smooth, then add the powdered sugar, milk, and vanilla extract, mixing until creamy.
9. Allow the brownies to cool completely before spreading the frosting evenly over the top. Slice into squares and enjoy!

Remember, the key to perfect brownies is not to overbake them. Watch for the edges to set while the center remains soft.