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Toss the tomatoes with the olive oil, vinegar and thyme leaves and season with salt and pepper. Put in a small baking dish in one layer and roast until completely soft and the juices have evaporated, about 30 minutes. Let cool slightly. Leave the oven on.
Heat the canola oil in a 10-inch nonstick pan over high heat until it begins to shimmer. Add the sausage and cook until golden brown, and then transfer to a paper-towel-lined plate using a slotted spoon. Remove the fat from the pan.
Whisk the eggs in a large bowl until light and fluffy. Fold in the drained sausage, tomatoes, provolone and Romano cheeses, arugula and chopped parsley and sprinkle with salt and pepper. Season the ricotta with salt and pepper as well and set aside.
Heat the butter in the nonstick pan. Add the egg mixture and stir to evenly distribute the ingredients using a heatproof rubber spatula. Cook until the eggs set on the bottom and around the sides of the pan, and then transfer to the oven. Bake until the eggs are cooked through and the top is lightly golden brown, 20 to 25 minutes.
Loosen around the sides of the pan with a rubber spatula and flip onto a serving platter or cutting board. Top with the ricotta and drizzle with extra-virgin olive oil. Grind some black pepper over the top, and scatter the whole parsley leaves over all.