Season the chicken. Season the bite-size chicken pieces on all sides with the salt and pepper. Pat them dry with a paper towel first if they feel wet, so they brown instead of steaming.
Brown the chicken. Heat a large skillet over medium heat, then add the cooking oil and swirl to coat the bottom. Add the chicken pieces in a single layer and cook for 3 to 4 minutes per side, until golden brown and cooked through to 165°F inside. Remove the chicken to a plate and cover loosely with foil to keep warm.
Cook the mushrooms. Add the minced garlic to the same skillet and sauté for about 1 minute, until it softens and smells fragrant. Add the sliced mushrooms and cook for 5 to 7 minutes, stirring occasionally, until they have wilted, darkened slightly, and look glossy and tender.
Build the sauce. Pour the chicken broth into the skillet and stir to dissolve all the browned bits off the bottom. Add the half-and-half and bring it up to a gentle simmer, then turn the heat down to medium-low and simmer, stirring often, for about 5 minutes, until smooth and slightly reduced. Stir in the Parmesan until it has melted in. Taste and season with more salt and pepper if needed.
Finish and serve. Add the chicken back to the skillet, along with any juices from the plate. Spoon the sauce over the top so every piece is coated, and warm together over low heat for 2 to 3 minutes. Check that the chicken reaches 165°F inside before serving. Top with fresh parsley if desired and serve hot.