Line an 8x8 baking pan with parchment paper. Set aside. Preheat the oven to 350 F/180 C.
In a stand mixing bowl, combine the lemon zest and sugar by rubbing it together with your fingers.
Then add the flour, baking powder, and salt, and mix until combined.
Add the cubed butter and mix on low speed until it looks like sand.
Then mix in the oil, egg, milk, sour cream, and vanilla extract until just combined.
Pour the batter into the prepared baking pan.
Bake for 23-26 minutes, or until a toothpick inserted into the center comes out clean. Let it cool completely on a wire rack.
In a mixing bowl, beat the cold mascarpone with the powdered sugar until combined.
Then add the cold whipping cream, vanilla extract, and lemon zest.
Whisk on low speed until combined, then increase the speed to medium and beat until stiff peaks form. Do not overmix or the frosting will look clumpy and curdled.
Gently combine the chopped strawberries, raspberries, blackberries, and jam in a medium bowl.
Spoon the frosting over the top of the cooled cake to the edges, then add the berries on top. If you plan on serving it later, I recommend leaving the berries until right before serving. Otherwise the berry juice will spread into the frosting and it won't look as pretty.