Prepare the chicken - Transfer chicken breasts to a large mixing bowl. Drizzle with approximately 1 tablespoon of olive oil and season with salt, pepper, and Italian seasoning, mixing well to coat the entire surface of each chicken breast.
Cook the chicken - Heat a large skillet over medium-high heat. When the pan is hot, drizzle with approximately 1 tablespoon of olive oil and swirl to coat the bottom of the pan. Work in batches, if needed (it took me two rounds to cook the 6 chicken breasts), to cook the chicken until golden, approximately 5-6 minutes per side, (or until nearly cooked through (see notes)). Remove chicken from the skillet and set aside.
Make the Sauce - Return the skillet to medium heat. Add the butter and shallots and sauté for 2-3 minutes, stirring often. Add the minced garlic and stir continuously for 30 seconds more. Mix in the artichoke hearts, fresh spinach, and chicken broth. Mix well to combine. Cook until the spinach has just wilted. Add the half-and-half and bring just to a simmer. Stir in the mozzarella.
Return the chicken - Finally, return the chicken back to the skillet. Reduce heat if needed to prevent the half-and-half from boiling. Allow the chicken to simmer and finish cooking for an additional 5 minutes or so.
Garnish and serve - Sprinkle with the shredded Parmesan cheese and additional salt and pepper, to taste. Garnish with fresh chopped parsley, if desired.