Position one rack 6 inches below the broiler and another at the bottom of the oven; heat the broiler on high.
Arrange 3 of the 4 tomato halves cut side down on a small rimmed baking sheet lined with aluminum foil and broil on the top rack until they’re blistered and charred, about 5 minutes. Turn the tomatoes over and broil until charred, about 4 minutes more. Let cool.
Finely dice the uncooked tomato half and set aside.
Put a large cookie sheet on the bottom rack and heat the oven to 500°F.
In a blender, pulse the charred tomatoes, including the skin, with the vinegar and garlic until coarsely chopped. With the motor running, slowly drizzle the oil through the hole in the lid. Transfer to a small bowl, season with 1 tsp. salt and 1/2 tsp. pepper, and set aside.
Put the pizza dough on a lightly floured surface. Using a bench knife, divide the dough into 2 equal pieces. Roll them into balls and set one aside, covered with a clean, damp kitchen towel. Using a floured rolling pin, roll the other ball into a 12- to 13-inch round and sprinkle with flaky sea salt or more kosher salt.
Using a peel, transfer the round to the cookie sheet in the oven. (Don’t worry if it buckles or wrinkles; this will make for a more interesting shape.)
Bake until the dough begins to bubble and brown underneath, about 2 minutes. Flip and bake until golden-brown around the edges and bubbly, about 2 more minutes. Transfer to a cooling rack. Repeat with the second ball of dough.
In a large bowl, toss the mustard greens and diced fresh tomato with enough of the vinaigrette to coat. Season to taste with salt and pepper.
Divide the greens between the flatbreads, spreading to cover, and serve.