Commence by bringing a sizable pot of water, amply salted, to a rolling boil. Subsequently, introduce the spaghetti into the boiling water. Permit the spaghetti to cook for approximately 10 minutes, ensuring it remains slightly firmer than al dente. Retain roughly ½ cup of the cooking water and then drain the noodles. Dress the noodles with one tablespoon of the allocated olive oil to prevent adhesion. Set aside for later use.

Prior to commencing the cooking process, ensure the oven is preheated to a temperature of 350°F (175°C). Take a 13x9-inch baking dish and prepare it by administering a light coating of nonstick spray.

In a robust Dutch oven or an equivalent large pot, heat the remaining tablespoon of olive oil over a medium-high flame. Following this, incorporate the diced onion and bell pepper into the pot, seasoning lightly with salt and black pepper. Allow the mixture to sauté for a duration of approximately 5 minutes or until the onions adopt a translucent appearance.

Introduce the diced tomatoes with chilies to the pot and maintain cooking until the majority of the liquid has been reduced, which typically takes an additional 5 minutes.

Carefully integrate the cream of mushroom soup, cream of chicken soup, and chicken stock into the mixture. Reduce the heat slightly to maintain a gentle simmer.

Add both the cheddar and Velveeta cheeses to the mixture, ensuring consistent stirring to facilitate even melting. Once a smooth sauce is achieved, introduce the prepared chicken to the mixture. Taste and season further with salt and pepper according to preference.

Incorporate the previously set-aside spaghetti into the sauce, ensuring it is evenly coated. If the consistency appears overly thick, judiciously add some of the retained pasta water to achieve the desired consistency.

Transfer the entire mixture to the pre-prepared baking dish, ensuring an even distribution.

Place the dish in the preheated oven and allow it to bake for a duration of 30-40 minutes. The dish is adequately baked when the sauce displays a bubbly nature and the surface possesses a golden hue. It is advised to serve the dish while still warm.