In large skillet over medium heat, cook bacon, stirring occasionally, until crispy, about 10 minutes; transfer to a plate. In medium bowl, whisk 3 cups cheese and the eggs. Meanwhile, cook spaghetti in a large pot of boiling, salted water until al dente, adding peas during the last minute of cooking. Save 1 cup of pasta cooking water, then drain and return pasta and peas to pot. Immediately whisk 1/2 cup pasta water into egg mixture; add to the pasta along with bacon. Toss until sauce coats pasta, adding more cooking water if dry. Season.