Preheat your oven to 450 degrees F. Put your pizza stone in the oven if you have one.
Roll out your pizza dough on a floured sheet of parchment paper to about 12 inches. (If you don't have a pizza stone, you should probably transfer the dough to a baking sheet now.)
Drizzle 1 tablespoon of olive oil over the top of the dough and use your fingers or a pastry brush to coat well, especially the edges.
Use a fork to mash a half cup of blackberries in a bowl. It doesn't have to be perfect. Use the fork or a slotted spoon to ladle the pulp onto the pizza. Spread it around like you would pizza sauce. (You don't need the remaining juice, but don't throw it out. Drink that sweet nectar if you know what's good for you.)
Top the smashed berries with 1 cup Parmesan and 1 cup swiss or mozzarella, making sure to get the crust.
Use a spoon to dollop the ricotta over the pizza. It doesn't have to be perfect.
Top with whole fresh blackberries and chopped green onions.
Sprinkle with salt, and add pepper to taste.
When the oven is hot, transfer the pizza to the oven. I transfer it using a a flat baking sheet, still on the parchment paper.
Bake for about 8-10 minutes, or until the top has started to brown. You can broil it for a minute if you want it crispier on top.
Remove from the oven and immediately sprinkle some chopped basil on top. Slice and enjoy!