Preheat your oven to 350°F (175°C). Grease an 8x8-inch baking pan or line with parchment paper. In a large bowl, prepare the brownie mix according to package directions (typically combining mix with eggs, oil, and water). Gently fold in the finely chopped fresh strawberries and strawberry jam until just combined. Do not overmix.

Pour the batter into the prepared pan and spread evenly. Bake for 28-30 minutes, or until a toothpick inserted into the center comes out with a few moist crumbs (not wet batter). The edges should be set. Allow the brownies to cool completely in the pan on a wire rack.

While the brownies cool, prepare the icing. In a small bowl, whisk together the powdered sugar, strawberry puree, milk, and vanilla extract until smooth. The icing should be thick but pourable. Add more milk a teaspoon at a time if needed.

Once the brownies are completely cool, pour the strawberry icing over the top and spread evenly with a spatula. Let the icing set for about 10 minutes before cutting into squares. For clean cuts, use a sharp knife wiped clean between slices.