Combine the 3 large eggs, 2 cups unbleached all-purpose flour, and a pinch of salt in the bowl of a mixer fitted with the paddle or dough hook.
Mix slowly, adding water a little at a time until the dough comes together — not dry and not sticky; adjust with more flour or water if needed.
Remove the dough, form into a ball, place on a cutting board and cover with a bowl; let rest for 10 minutes.
Run the rested dough through a pasta machine following the manufacturer's instructions, finishing on a wide setting (about a #6 on KitchenAid) to make long pasta strips.
Bring a large pot of salted water to a boil, then cook the spaghetti rigati until al dente according to thickness (test frequently).
While the pasta cooks, heat the tomato sauce in a skillet or saucepan and cook the chicken in the sauce until the chicken is fully cooked through.
Toss the drained pasta with the chicken and tomato sauce and serve immediately.
