Slice the red peppers in half, remove the stem and seeds and plate on a baking tray over some parchment paper. Bake in the oven for 50 minutes, or until the pepper's skin starts to blacken and can be peeled. Take the peppers out of the oven and let cool down. Once cooled, remove the skin with your hands.
Add the peeled peppers to a food processor (or a blender, stick blender) and mix until smooth.
Sprinkle the salt and pepper over the chicken breasts on each side. Mince the garlic, slice the mushrooms, shred the spinach and slice the fresh basil leaves.
Heat the olive oil in a large cast iron skillet and place the breasts in. Cook on each side for 4-5 minutes on medium heat. Once each side is grilled, add the garlic and mushrooms and cook for a minute. Add the pureed red peppers and 1/2 cup of water. Cover and let simmer on medium-low heat for 5 minutes.
Uncover, add the cream and the spinach. Move the spinach around with your spatula until they are completely wilted. Cook for a minute, just so that the cream and red peppers combine and turn off heat. Crumble the goat cheese over and top with the sliced basil leaves.