Add potatoes to a large stockpot and cover with water. Heat over high heat until the water reaches a boil. Reduce heat to medium-high and continue boiling until the potatoes are fork-tender (can mash easily with a fork). Drain, and then return the potatoes to the stockpot.
Meanwhile, while the potatoes are cooking, heat 1 tablespoon butter in a large saute pan over medium-high heat until melted. Add onion and saute for 5 minutes, or until soft and translucent, stirring occasionally. Stir in green bell pepper, red bell pepper, and jalapeno, and continue cooking for another 4-5 minutes or until the peppers are soft and cooked through, stirring occasionally. Stir in the green chiles. Remove pan from heat and set aside.
When the potatoes are cooked and ready, add in the onion mixture, cheese, milk, salt, pepper, and the remaining 3 tablespoons of butter (cut into small pieces) to the stockpot. Then use a potato masher to mash the potatoes and stir the mixture together until combined. Taste, and season with additional salt and pepper if needed, and feel free to add in extra warmed milk if you would like creamier potatoes.
Serve warm, garnished with chopped green onions and/or extra shredded cheese if desired.