Preheat oven to 375F.
Coat the beets with a drizzle of EVOO and place them on a roasting pan. Cover the pan with foil and place in the oven. Roast the beets for about 30 minutes if using baby or about 45 minutes if using medium, then remove from the oven and let cool for 15 minutes, until cool enough to handle. (If using red beets, coat your hands with a little oil before peeling them to prevent them from staining your hands.)
Just before the beets come out of the oven, bring a large pot of water to a boil for the pasta. Once boiling, salt the water liberally, add the pasta, and cook to al dente. Heads up: reserve 1 cup of the starchy cooking water before you drain the pasta.
While the beets cool, heat 3 tablespoons of the EVOO, about three turns of the pan, in a skillet over medium heat. Add the shallots and garlic, and saut a few minutes until they start to soften. Add the peas, gently cook for 3 or 4 minutes, until heated through then season with salt and freshly ground black pepper.
Peel the beets and add them to a food processor with the ricotta cheese. Puree for a minute or so, until combined into a smooth, golden mixture. Transfer to a large pasta bowl and season with about 2 teaspoons of lemon zest, salt, freshly ground black pepper, the parsley and 1/2 cup of grated Parmigiano. Add the reserved starchy cooking water, the hot pasta, and the peas and shallots. Toss to combine and let sit for a minute to allow the pasta time to absorb the flavors. Just before serving, give it another quick toss.
In a bowl, add the arugula and the tomatoes. Toss the salad with lemon juice to taste, the remaining 3 tablespoons of EVOO, salt and freshly ground black pepper. Pass additional Parmigiano for the pasta at the table and the salad alongside.