Preheat the oven to 200˚ F. Heat 1 tablespoon olive oil in a large nonstick skillet over medium-high heat. Add the onion, tomatoes and chard stems and cook until the vegetables are softened, about 5 minutes. Add the chard leaves and cook, stirring, until wilted, about 3 minutes; season with salt and pepper. Transfer to an ovenproof dish and keep warm in the oven. Wipe out the skillet and return to the stovetop.
Whisk 3 eggs, the milk, sage, 1 teaspoon pepper and 1/2 teaspoon salt in a baking dish. Lay the bread in the dish, flipping the slices once or twice to fully soak. Heat 1 tablespoon olive oil in the skillet over medium heat. Carefully lift the bread from the egg mixture, letting the excess drip off. Cook in the skillet until golden, about 3 minutes per side. Transfer to a baking sheet, top each with a slice of cheese and melt in the oven.
Wipe out the skillet and heat the remaining 1 tablespoon olive oil over medium-high heat. Crack in the remaining 4 eggs and cook until the whites are set, about 5 minutes. Divide the French toast among plates. Top each with the vegetables and a fried egg; season with salt and pepper. Drizzle with the hot honey.