Bring a large pot of water to a boil. Add chicken, 1/2 teaspoon salt, and 1/2 teaspoon black pepper; cover and reduce heat to a gentle boil. Cook until chicken is no longer pink in centers and juices run clear, 10 to 12 minutes. An instant-read thermometer inserted into centers should read at least 165 degrees F (74 degrees C). Remove chicken; set aside until cool enough to handle, then chop into bite-sized pieces. Leave cooking liquid in the pot.

Bring liquid back to a boil. Add spaghetti; cook, stirring occasionally, until tender yet firm to the bite, about 12 minutes. Drain; do not rinse. Discard liquid.

Melt butter in the same pot. Add onion and bell pepper; cook until softened, 5 to 7 minutes. Add chicken soup, mushroom soup, diced tomatoes and chiles, chicken, and spaghetti, one ingredient at a time, gently stirring after each addition.

Add Mexican cheese blend; stir to combine. Season with salt and black pepper. Cook until cheese is thoroughly melted, stirring occasionally to keep from scorching.