Position a rack in the center of the oven and heat the oven to 425°F. Butter a large rimmed baking sheet.
In a 9x13-inch baking dish, whisk together the eggs, Parmigiano, milk, thyme, Tabasco, 1/4 tsp. salt, and 1/8 tsp. pepper. Put the bread in the dish in a single layer, turn to coat on both sides, and let sit, turning once more, for 5 minutes.
Arrange the bread on the baking sheet. With your fingers, gather any remaining cheese from the baking dish and spread it over the slices. Bake until the bottoms are golden, 8 to 10 minutes. Flip and continue baking until the bottoms are golden, about 5 minutes.
Meanwhile, in a large bowl, combine the lemon juice, honey, 1/4 tsp. salt, and a few grinds of pepper; whisk in the olive oil. Add the arugula and toss.
Serve the French toast with the salad and tapenade on the side.