In a large saute pan, heat the oil until it is just smoking. Add the onion, honey, and 2 cups of the peas, and saute until softened and cooked through, about 10 to 12 minutes.
Place peas in a food processor and pulse until coarsely pureed, season generously with salt and pepper, and set aside.
Bring 6 quarts of water to a boil and add 2 tablespoons salt. Melt the butter in the saute pan, add the remaining peas, and cook slowly until just softened, 4 to 5 minutes. Add the pea puree to the whole peas and set aside.
Just before the pasta is done, pour a ladle of the starchy cooking water into the pan with the pea puree and stir to loosen the sauce. Cook the pasta until al dente and drain well, reserving more of the pasta water. Immediately toss the pasta into the pan with the pea mixture and place over medium heat. Stir gently to mix well, adding a little pasta water to achieve the correct texture, not too dry and not too wet - the noodles should be dressed like the greens of a salad. Add the cheese and mint leaves, and toss to combine. Serve immediately.