Bring 3 cups of chicken stock to a boil. Whisk the gravy mix into the remaining 1 cup of stock until lump-free. Pour slurry into boiling stock and whisk to fully incorporate. Bring back to a boil, reduce heat, and simmer for 1 minute. Remove from heat.
Remove stems and excess gills from mushrooms. Chop stems and any pieces removed.
Sauté chopped mushroom stems and onions in butter over high heat until browned. Turnoff heat and add garlic and red pepper flakes, stirring to evenly distribute throughout mixture. Add mixture to prepared gravy.
Cook green beans in boiling salt water for 5–7 minutes. Drain. Add salt and black pepper to taste. Transfer to a large mixing bowl. Pour gravy mixture over beans and stir to evenly coat all pieces. Set aside.
Brush mushroom caps with melted butter and place gill side down on parchment-lined baking sheet; season with salt. Bake in a convection oven at 400˚F for 5 minutes.
Remove from oven and let cool slightly. Transfer to another parchment-lined baking sheet, gill side facing up. Use a paper towel to remove any excess liquid. Fill each cap with gravy/bean mixture. Top each filled cap with 1 oz. Parmesan cheese.
Return to oven and bake for another 5 minutes. Garnish with 1 oz. french fried onion per mushroom.