When chicken comes out of the oven, transfer it to a plate or cutting board to rest and return the skillet to the stovetop over medium heat. Add the pancetta to the skillet the chicken was cooked in and render until crispy and golden, 2-3 minutes. Add the pepper flakes and chopped garlic and cook until aromatic, about 30 seconds. Add wine to the pan to deglaze, then the chicken stock, scraping up any bits that are stuck to the bottom. Reduce for about 2-3 minutes.
Place the egg yolk into a small bowl and whisk lightly with a fork. Add a large spoonful of the hot liquid from the pan to the bowl of yolks while stirring then add the yolk mixture back to the pan and stir over low heat until slightly thickened, being careful not to scramble the egg yolk. Remove the skillet from the heat and add in the grated cheese and parsley. Reserve the sauce warm while the escarole cooks.
Place a large skillet over medium-high heat with 1 turn of the pan of EVOO, about 1 tablespoon. Add the crushed garlic and anchovy filets and remaining crushed red pepper flakes to the skillet and cook until anchovies are paste-like and aromatic, about 30 seconds. Add the escarole leaves to the skillet and cook until wilted and tender, about 3 minutes. Season the greens with salt and pepper.
To serve, arrange some of the greens on each dinner plate, rest the chicken on top and pour some of the carbonara sauce over everything.