Brush oil over the bottom and up the sides of a 12”, heavy, ovenproof skillet.
Halve the tomatoes lengthwise and remove seeds, being careful to keep halves intact.
Whisk the butter with the vinegar and corn syrup; toss in the tomatoes and ensure all are coated. Arrange tomatoes cut-side-up so they fit snuggly in the skillet. Sprinkle with salt and pepper. Roast for 1 hour.
Meanwhile, pulse the flour, butter, and ½ cup of Black Diamond Shredded Italiano cheese in a food processor until uniformly crumbly. Continue pulsing while drizzling in the ice water until the pastry comes together in a ball; wrap and chill for 15 minutes. Roll the pastry out on a lightly floured surface into an 11” circle. Chill until ready to use.
Remove skillet from the oven. Sprinkle remaining cheese evenly over the tomatoes. Carefully arrange the dough over the tomatoes; folding back as needed so it doesn't touch the sides of the skillet. Bake for 30 minutes or until pastry is golden.
Remove from oven and let cool for 5 minutes. Place a large platter over the skillet and then, with oven mitts on, flip the tarte onto the platter.
Garnish with additional cheese and fresh basil leaves.