Combine the 3 large eggs, 2 cups unbleached all-purpose flour, and a pinch of salt in the bowl of a stand mixer fitted with the dough hook (or use a mixing bowl and fork).
Mix slowly while adding water a little at a time until a cohesive dough forms; it should not be flaky or overly wet. If too sticky, add a little more flour.
Remove the dough, place it on a lightly floured cutting board, cover with a bowl, and let rest for 10 minutes.
Roll the rested dough into a sheet and pass through a pasta machine following the manufacturer's directions until you reach a medium-thin setting (about setting 6 on a KitchenAid), then cut into spaghetti rigati.
Bring a large pot of salted water to a boil and cook the spaghetti rigati until al dente, then drain, reserving some pasta water if desired.
In a skillet, cook the chicken in the tomato sauce over medium heat until the chicken is fully cooked through and the sauce is hot and combined.
Toss the cooked spaghetti with the chicken and tomato sauce, adjust seasoning, and serve immediately.