In a large cast-iron skillet or brazier, cook the bacon over medium-high heat, stirring often, until the fat has rendered and the bacon is crisp, 8 to 10 minutes. Transfer the bacon to a medium bowl using a slotted spoon, reserving the drippings in the skillet.

Arrange the mushrooms in the skillet in a single layer and cook, stirring every 2 minutes and re-spreading into an even layer, until lightly golden brown, about 6 minutes. Transfer to the bowl with the bacon. Do not wipe the skillet clean.

Sprinkle the chicken evenly with the salt and pepper. Heat the oil in the same skillet over medium-high heat. Cook the chicken, flipping once, until golden brown on both sides, about 6 minutes per side. Remove from the heat.

Spoon about 1 heaping tablespoon of the honey mustard on top of each chicken breast in the skillet, then top each evenly with the mushroom-bacon mixture. Sprinkle evenly with the Monterey Jack and cheddar cheese.

Place the skillet over medium-low heat. Cover with a tight-fitting lid or aluminum foil. Cook, undisturbed, until the cheese is melted and an instant-read thermometer inserted into the thickest part of the chicken registers 165°F, 8 to 10 minutes.