Rosita Missoni loves pan-frying scamorza, a firm Italian cow's-milk cheese, to give it an irresistible crust. Provolone would also work.
Preheat the oven to 325°. On a rimmed baking sheet, arrange the tomatoes in a single layer. Drizzle with olive oil, season with salt and sugar and top with the oregano. Bake for 1 hour, until the tomatoes are soft but still hold their shape. Discard the oregano.
Meanwhile, in a food processor, puree 3 cups of the arugula with 1/4 cup of the olive oil until smooth. Season with salt and scrape into a bowl. Wipe out the food processor.
Add the sun-dried tomatoes and 1/2 cup of the olive oil to the processor; process to a coarse puree and season with salt.
In a bowl, whisk the vinegars with the vegetable oil and 1 1/2 tablespoons of the olive oil. Season with salt.
Heat a nonstick skillet. Pan-fry the cheese over high heat, turning once, until well-browned and just starting to melt, 2 minutes. Transfer the scamorza to plates.
In a large bowl, toss the remaining 9 cups of arugula with the dressing. Mound the salad on the plates alongside the cheese. Dot the cheese with the pestos. Arrange the tomato slices around the salad, drizzle with olive oil and serve right away.