Preheat the oven to 350°F (175°C). Line a baking sheet with parchment paper.
Season the chicken fillets with most of the chicken bouillon and garlic salt.
Heat 1 tablespoon of oil and the butter in a cast-iron skillet over medium-high heat.
Sear the chicken fillets for about 3 minutes on each side until they are golden brown.
Remove the chicken from the skillet and place it on the prepared baking sheet.
Add the remaining oil to the skillet. Add the onions and mushrooms, and sauté until tender, about 5-7 minutes.
Season with the remaining chicken bouillon and garlic salt.
Pour in the cooking wine and stir, scraping up any browned bits from the bottom of the skillet.
Spoon the onion and mushroom mixture evenly over the chicken fillets.
Sprinkle the shredded Monterrey Jack cheese on top.
Transfer the baking sheet to the preheated oven and bake for 15 minutes, or until the chicken reaches an internal temperature of 165°F (74°C).
Serve the smothered chicken hot, garnished with fresh herbs if desired.