Bring a pot of salted water to boil. Add pasta and cook until al dente.
Sauté garlic and anchovy paste in oil in a large skillet over medium-low heat, until garlic is lightly browned.
Add 5 eggs gently to the oil, and cook until the whites are set, but the yolks are still soft (“over easy”).
Reserve 1 cup of pasta water before draining pasta.
In a large serving bowl, toss cooked pasta with eggs and oil mixture (when tossing, the eggs will break up and form a creamy sauce). Add 2 ladles of the reserved pasta water to the bowl to help make the sauce (the eggs will finish cooking in the heat of the pasta).
Add the sliced cherry tomatoes and baby spinach to the bowl, gently combining until spinach wilts. Spinach will wilt from the heat of the pasta, but you may need to add 1 more ladle of pasta water as the ingredients will absorb the sauce.
Serve immediately, sprinkling individual bowls with grated Parmesan cheese.