Combine the yeast and water in a large mixing bowl of an electric mixer fitted with a dough hook. Whisk until the yeast is dissolved. Mix in 3 tablespoons of the olive oil, the basil, garlic, flour, cornmeal, and 1/2 tablespoon of the kosher salt. With the mixer on medium speed, mix until the dough pulls away from the sides of the bowl and forms a ball. Turn the dough out onto a floured surface and knead until it forms a smooth ball.
Lightly oil the inside of another large mixing bowl. Place the dough in the bowl and turn to oil all sides. Cover the bowl with a clean kitchen cloth and put it in a warm, draft-free place. Let rise until it doubles in size, about 1 1/2 hours.
Preheat the grill to medium heat.
Remove the dough from the bowl, invert it onto a heavily floured work surface, and punch it down. Divide the dough into 4 equal portions and form each into a ball. With your fingers, press each ball into a round about 1/4 inch thick. Brush the rounds with the remaining 1 tablespoon olive oil. Place on the hot grill until cooked all the way through, 3 to 4 minutes on each side. Remove and sprinkle with the remaining 1/2 tablespoon kosher salt.
In a large mixing bowl, toss the salad greens with 1/4 cup of the aioli dressing. Season with salt and pepper. Season the tomato slices with salt and pepper.
To serve, spoon the remaining dressing in the center of each plate and on the rim. Slice the flat breads in half crosswise.
For each sandwich, tuck in 2 slices of tomato, 3 slices of bacon, 2 slices of cheese, and some of the lettuce and place on top of the dressing and sprinkle with parsley.
Yield: 4 servings