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Preparation Preheat oven to 350°F. Toss onion, soy sauce, garlic powder, and 1 Tbsp. oil on a large rimmed baking sheet to coat. Season with salt. Bake, tossing once halfway through, until onion is browned around the edges and very tender, 25–35 minutes. Let cool 5 minutes.
Meanwhile, toss herbs, hot sauce, and remaining 1 Tbsp. oil in a small bowl. Season with salt, if needed.
Melt butter in a small nonstick skillet over medium heat. Add eggs; season with salt. Cook, stirring with a rubber spatula, until eggs have set into creamy folds, 1–2 minutes.
Spread herb sauce on cut sides of muffins. Transfer bottom halves to a rimmed baking sheet. Top with eggs, then lay cheese over. Bake bottom halves just until cheese is melted, about 5 minutes. Top with onion and close with muffin tops.
Do Ahead: Onion can be cooked 5 days ahead. Transfer to an airtight container and chill. To make a sandwich quickly without turning your oven on, fold the cheese into the eggs when they have just begun to set, top with onion (no need to rewarm), and assemble with toasted English muffins.