Bring water and olive oil to a boil in a stockpot over high heat. Break pasta in half and add to water. Cook until pasta is al dente, 7 to 10 minutes.
Cook bacon in a large skillet over medium heat until fat is rendered, 5 to 7 minutes; do not drain fat. Add mushrooms and scallions; toss to combine.
Strain pasta; add to bacon mixture. Increase heat to high; toss until pasta is well-coated. Crack eggs into skillet; toss until eggs are cooked through, about 3 minutes. Add Mizithra cheese, mozzarella cheese, and salt; toss just until combined.