►Preheat oven to 350°F.
► Combine eggs and heavy cream in medium bowl; whisk to thoroughly blend. Season with salt and pepper.
► Place egg and cream mixture in a 9 x 13-inch pan and add bread slices. Leave bread to soak in mixture for 15 minutes; flip slices and soak for additional 10 to 15 minutes.
► As bread soaks, place large nonstick skillet over medium heat. Add bacon slices and cook until crisp; remove and drain on paper towel-lined plate. Set aside.
► Remove most of the bacon drippings from skillet and add oil. Working in batches, add soaked brioche slices and cook on each side until golden-brown.
► Transfer slices to parchment-lined baking sheet. Bake in preheated oven until custard is cooked through, 6 to 8 minutes (the bread should soufflé, or puff up).
► Remove pan from oven, sprinkle grated cheese over each slice, and return to oven; bake until cheese melts, about 4 minutes.
► Remove pan from oven. Place 1 slice of French toast on each serving plate. Top each with 2 bacon slices and 2 tomato slices.
► Place greens in a large bowl and toss with olive oil. Place greens on top of each French toast. Serve warm.