Preheat the oven to 350°F (180°C).
Butter or spray a square 8-inch baking pan or cake pan. If you use a 9-inch pan, the cake will be shorter and bake faster, so check for doneness earlier.
If you want, line the pan with a piece of parchment paper, covering its bottom and two sides. The other two sides will remain unlined but greased. This will help you remove the cake after it's baked.
In a large bowl or the bowl of an electric mixer, beat ½ cup unsalted butter with ½ cup sugar for 2 minutes at medium-high speed.
Add 2 eggs, one at a time, and beat until well incorporated, scraping the sides of the bowl once or twice.
Add 1 teaspoon vanilla extract and 2 teaspoons lemon zest, and mix well.
Add 1 ¼ cups whole milk ricotta cheese and combine everything. I use the spatula from this step onward to avoid overmixing the batter. Sift 1½ cups all-purpose flour, 2 teaspoons baking powder, and ½ teaspoon salt (I do it directly over the mixture, but you can use a separate bowl) and add to the wet ingredients in three parts, alternating with ¾ cup milk in two parts. That means that you start and end with flour. Do not beat much, just until everything is combined. The dough is very creamy but with texture.
Scrape batter into the prepared pan and scatter 1 ⅓ cups berries on top.
Bake cake for about 30 minutes, until it has light golden brown sides and the center is almost set. If you use a cake tester or toothpick it should come out almost dry, or with very few moist crumbs attached. Don't overbake.
Let the cake cool completely on a metal wire rack before removing it from the pan. First, loosen the unlined sides with a smooth-bladed knife or spatula, and then lift the whole block (carefully because it's a soft cake) and transfer it to a board. Cut into squares and serve.
Refrigerate leftovers covered in plastic wrap or an airtight container.