Clean the chicken breasts, pat them dry with paper towels, and season both sides generously with salt and black pepper.
Heat a large skillet over medium-high heat, add a couple tablespoons of olive oil, and sear the chicken breasts on both sides until golden brown, then remove them from the pan and set aside.
Add a little more oil to the same pan, lower the heat, and sauté the minced garlic until lightly golden, then stir in the chopped onion with a pinch of salt and cook until soft, sweet, and almost caramelized.
Turn up the heat, toss in the sliced mushrooms with another small pinch of salt, and cook until they release their moisture and start to brown, then stir in the oregano, basil, and chopped sun-dried tomatoes and let everything cook together for a couple of minutes.
Pour in the white grape juice and vinegar mixture and let it simmer for about 2 minutes to mellow out, scraping the bottom of the pan to pull up all that flavor.
Add the chicken broth and cooking cream, stir everything together, and let it simmer on low for 5 minutes so the sauce starts to thicken.
Stir in the grated cheese until it melts smoothly into the sauce, then add the spinach and cook until it wilts down and blends in.
Nestle the chicken breasts back into the pan, spoon some sauce over the top, cover with a lid, and cook on low for about 10 minutes, flipping them halfway through.
Uncover, let the sauce bubble gently for another minute if it needs to thicken, and serve hot with extra black pepper on top.