Place 2 baking sheets in oven and preheat to 500 degrees F.
Peel cucumbers. Cut them in 1/2 to remove seeds and slice them widthwise into slices about 1/8-inch thick.
Core the tomatoes and dice into 1/2-inch pieces.
On a lightly floured surface, using a rolling pin, roll dough balls into a circle about 8 inches in diameter and about 1/8-inch thick. With a spoon divide the hummus evenly between the 4 dough rounds and spread thin to cover the entire surface.
Remove the baking sheets from oven and sprinkle evenly with cornmeal. Transfer the prepared dough rounds to the baking sheets and bake until slightly underdone (lightly browned but still pliable), about 8 to 12 minutes.
While the dough rounds are baking, toss together the cucumber slices, diced tomatoes, oregano, garlic, chili flakes, olive oil, and vinegar, season with gray salt and pepper, to taste.
Let the baked dough cool slightly then top with the cucumber salad divided evenly among the 4 doughs. Crumble feta cheese over the top and serve open faced.