Melt the unsalted butter in a large mixing bowl. Add granulated sugar and brown sugar, stirring until well combined. Add eggs one at a time, mixing thoroughly after each addition. Finally, add vanilla extract.
In a separate bowl, beat the softened cream cheese until smooth. Gradually add powdered sugar and the second teaspoon of vanilla extract. Mix until well blended, then gently fold in sliced strawberries and lemon juice.
Once the brownie crust has cooled completely, pour the cheesecake filling over it and spread evenly with a spatula.
Bake in a preheated oven at 350°F for 20-25 minutes. A toothpick inserted in the center should come out with a few moist crumbs.
After baking, let the pan cool completely. Cover and refrigerate for at least 120 minutes before serving.