Slice the tomatoes place into a bowl or large plate. Tear or chop the basil leaves.
Drizzle the tomatoes with the olive oil, add the basil and gently toss to coat. Set aside.
Preheat the oven to 475 degrees Place 2 baking sheets in the oven.
In a separate bowl whisk the flour and salt together. Add the water and mix until a smooth dough forms.
Separate the dough into 4 equal pieces (use a kitchen scale.)
On a lightly floured pieces of parchment paper, roll each piece of dough out into an oval shape.
Beat the cream cheese and sour cream together until smooth and combined. Season with Kosher salt and freshly ground pepper to taste.
Spread the mixture out evenly over the four dough ovals.
Lift up the parchment paper and transfer the Flammkuchen on the parchment paper onto the preheated baking sheets in the oven.
Bake for 15 minutes until the crust is golden brown and crispy.
Remove from oven. Season again with salt and freshly ground pepper, if desired. Serve warm.