Heat 1 Tablespoon oil in large non-stick skillet over medium-high heat.
Season chicken with ½ tsp. thyme, kosher salt and pepper and cook until browned, turning once, and set aside.
In same skillet melt butter and add remaining 1 Tablespoon oil.
Cook onions over low heat until golden and softened, stirring occasionally, about 25 minutes.
Add sherry and cook 1 minute.
Combine broth, flour and remaining ½ tsp. thyme until blended and stir into onions.
Return chicken to skillet and bring to a boil.
Reduce heat and simmer, uncovered, until chicken is cooked through and a sauce has formed.
Top with shredded cheese. Reduce heat to low and cook, covered, until cheese is melted, about 2 minutes.