Cube potatoes; slice shallots and bell peppers. Combine on large cookie sheet with kosher salt, black pepper, butter and olive oil. Mix with hands to distribute herbs and oil/butter evenly.
When done, place in oven, roasting anywhere from 45 minutes to 1 hour, depending on how you prefer the tenderness of your vegetables.
Approximately 20 minutes before vegetables are finished roasting, heat 1/8 cup olive oil in frying pan. When hot, put in chicken breasts (I keep the breasts whole and then cube when still hot, but you can cube ahead of time and then saute in oil, if you prefer), cooking approximately 4 - 8 minutes per side, depending on thickness of breast. (Make sure not to overcook, which will make the chicken dry.) Remove chicken from oil a few moments before fully done, as it will finish cooking with the roasted vegetables.
Add chicken to roasted vegetable mixture, mixing well. You should have about ten minutes left of roasting time. You may like to add a bit more olive oil to the entire mixture at this point if you think it is needed (for example, if the vegetables look dry).
Finish roasting until vegetables are at desired tenderness, then remove from oven.
Garnish with Gorgonzola cheese (or bleu cheese, or your favorite cheese) and serve.
Note: Finesse is needed with the chicken. If it cooks too long with the roasted vegetables it may dry; if it cooks too little in the oil before being added to vegetables, it may not cook all the way. Watch carefully.