If using fresh English peas in the pod, shell them (you should get a little more than 1 cup.) Remove and discard the strings from the snow peas and cut them into slivers. Trim and thinly slice the spring onions; peel and thinly slice the garlic. Set all prepped vegetables aside.
Peel the onion and cut it in half from stem to stem; trim off stem ends then slice very thinly. In a large skillet combine the onion, thyme and 2 tablespoons olive oil. Bring to a sizzle, season with salt and pepper then reduce the heat to low and slowly caramelize until softened and deeply golden brown, about 20 minutes. While the onions cook, thinly slice the garlic and spring onions. Trim the strings and ends from the snow peas and sliver them lengthwise. Set aside.
In a serving bowl whisk together the eggs and cream until fully blended, then stir in the two cheeses. Add ½ teaspoon each salt and freshly ground black pepper. Bring a large pot of water to a boil and add two big handfuls of salt—you want the water to taste like the sea.
When the onions are caramelized and golden, add the pasta to the boiling water and stir; bring back to the boil and cook according to the instructions on the box.
While the pasta cooks, raise the heat under the caramelized onions and add the butter to the skillet. Stir in the sliced garlic and spring onions; sauté, stirring constantly, until fragrant and beginning to brown slightly. Add the slivered snow peas and shelled (or defrosted) peas and season with salt and pepper; sauté until just bright green, no more than 3–4 minutes. Remove from the heat. Set aside.
As soon as the pasta is al dente, reserve 1 cup cooking water in a measuring cup and drain the rest. Whisking constantly, dribble in half the cooking water into the egg- cheese mixture, then add the drained pasta and toss to combine. Add the peas and onions to the bowl along with another ¼ cup pasta water and continue to toss until the pasta and peas are fully coated and the sauce is very creamy. Add more cracked black pepper to taste, then serve immediately with extra Parmigiano grated over the top of each bowl.