Fill a 12-quart stock pot 2/3 full with water. Bring the water to a boil.
Add one teaspoon of the salt.
Carefully place the chicken in the boiling water. Reduce the heat to medium and boil until chicken is cooked through, about 20 minutes. Remove the chicken from the pot and set aside to cool, but don’t discard the water. Once the chicken is cool enough to handle, pull it apart and shred it.
Bring the water back to a boil. Add the remaining teaspoon of salt. Add the spaghetti and cook the noodles according to package directions. Before draining the pasta, ladle out one cup of the cooking water and set aside. Drain the pasta.
In the same pot, melt the butter over medium heat. Sauté the bell pepper and onion until tender, about 5-7 minutes.
Chop the Velveeta into smaller pieces and add it to the pot. Pour in the Rotel tomatoes with juice. Add the pasta and shredded chicken. Stir everything to combine. If the mixture seems too thick, add a little bit of the reserved cooking water to thin it out.
Preheat the oven to 350 degrees.
Spread the pasta mixture into the bottom of one 9 x 13 x 2-inch casserole dish or two 8 x 8-inch casserole dishes.
Top the chicken spaghetti with the shredded cheese. Bake for 20-30 minutes until the cheese is melted and the casserole is cooked through.