Preheat the oven to 350°F (175°C). Line a 9 × 13-inch baking pan with parchment paper or coat it evenly with nonstick cooking spray. Properly preparing the pan will prevent sticking and simplify removal after baking.
Place the softened butter, granulated sugar, and packed brown sugar in a large mixing bowl. Blend until the ingredients are evenly combined.
Add the quick oats, all-purpose flour, baking soda, and salt. Stir only until a cohesive, crumbly mixture forms.
Transfer half of the oatmeal mixture to the prepared pan. Press it firmly and evenly across the bottom to form the crust.
Bake the crust for 13 minutes.
While the crust is baking, place the cream cheese in a separate large bowl. Beat or stir with a spatula until completely smooth.
Gradually incorporate the sweetened condensed milk, mixing continuously to maintain an even, creamy consistency.
Add the vanilla extract and egg yolk. Mix thoroughly until the filling is smooth and fully blended.
Remove the partially baked crust from the oven. Pour the cream cheese filling over the hot base and spread it into an even layer.
Distribute the blueberries or raspberries uniformly over the filling.
Scatter the reserved oatmeal crumble across the fruit. Press the topping down gently so it adheres without compressing the filling.
Return the assembled pan to the oven and bake for an additional 20–25 minutes, or until the outer edges are golden brown.
The center should appear set while retaining a slight jiggle. Avoid overbaking, as this may cause the cream cheese layer to become dry rather than creamy.
Allow the bars to cool completely in the pan. Once cooled, refrigerate them for a few hours so the filling can become firm.
Cut the chilled dessert into squares and serve cold or lightly chilled.