Beat eggs together with sage and Pecorino.
In a nonstick saute pan, heat 2 tablespoons olive oil over medium heat. Pour in egg mixture and cook until set like an omelette, flipping once.
Meanwhile, in a 12 inch to 14 inch saute pan, heat remaining oil until smoking. Add scallions and celery and saute until just softened, about 4 to 5 minutes. Add basic tomato sauce and simmer 5 to 6 minutes. Cut salivate into wedges and add spoonfuls of sauce. Serve immediately.
Saute the onion and garlic in the olive oil over medium heat until translucent, but not brown (about 10 minutes). Add the thyme and carrot and cook 5 minutes more. Add the tomatoes. Bring to a boil, lower the heat to just bubbling, stirring occasionally for 30 minutes. Season with salt to taste. Serve immediately, or set aside for further use. The sauce may be refrigerated for up to one week or frozen for up to 6 months.