Melt butter in a large skillet over medium-high heat; add onion, and cook, stirring often, 6 to 8 minutes or until tender. Add mushrooms, and cook, stirring often, 5 minutes. Reduce heat to medium, and cook 5 more minutes or until mushrooms are tender.
Bring whipping cream to a boil in a medium saucepan over medium heat. Reduce heat to low, and simmer, stirring often, 15 minutes or until slightly thickened.
Whisk crumbled cheese into whipping cream, and cook, whisking often, over medium heat 4 minutes or until cheese is melted. Stir in onion mixture. Keep warm.
Sprinkle fillets with salt and pepper. Grill fillets, covered with grill lid, over medium-high heat (350° to 400°) 5 to 8 minutes on each side or to desired degree of doneness. Remove from grill, and let stand 10 minutes. Serve with warm cheese sauce. Garnish, if desired.
Note: To make recipe on an indoor grill pan, prepare as directed through Step Heat a nonstick grill pan over medium-high heat. Sprinkle fillets with salt and pepper. Cook fillets 4 to 5 minutes on each side or to desired degree of doneness. Proceed with recipe as directed.