Preheat the oven to 475°. In a 12-inch ovenproof nonstick skillet, melt the butter in the olive oil. Add the onions, thyme, crushed red pepper, bay leaf and a generous pinch each of salt and pepper. Cover and cook over moderate heat, stirring occasionally, until the onions are very soft and browned, about 20 minutes. Discard the bay leaf.
In a large bowl, whisk the eggs with the crumbled goat cheese, Parmigiano, parsley, chives, 1/2 cup of the Gruyère, 1 teaspoon of salt and 1/4 teaspoon of pepper. Pour the egg mixture into the skillet and cook over moderately low heat, stirring gently, until the eggs start to set, about 5 minutes. Sprinkle the frittata with the remaining 1/2 cup of Gruyère. Transfer the skillet to the oven and bake the frittata for 8 to 10 minutes, until the center is just set. Run a rubber spatula around the edge of the frittata and slide it onto a work surface. Let cool slightly, then cut it into 6 pieces.
In a medium bowl, toss the arugula with the vinegar and season with salt and pepper. Brush the bagels with butter and fill with the frittata, arugula, sliced tomatoes, prosciutto and basil. Close the bagels, cut in half and serve.
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