Preheat oven to 400ºF.
Bring a large pot of water up to a boil, generously salt the water and cook the spaghetti to al dente, according to package directions. Before draining pasta, reserve about a cup of the starchy cooking liquid.
On a baking sheet, toss the tomatoes with one turn of the pan of EVOO, salt and pepper. Transfer to the oven and roast until tender and bursting, about 10 minutes.
While the tomatoes are roasting, start the pesto: In a food processor, add the arugula, basil, tarragon, pine nuts, lemon zest, garlic, salt and pepper. Pulse a few times, add the Parmigiano-Reggiano, then drizzle in the EVOO with the machine running until it's all combined.