Heat olive oil in a large skillet set over medium heat.
Season the chicken with salt, pepper, and Italian seasoning and add it to the heated oil.
Cook the chicken for about 6 minutes per side or until no longer pink. Remove from skillet and set aside.
Add the butter to the skillet and melt it over medium heat. Add the asparagus and onions and cook for 4 minutes or until the asparagus is crisp-tender.
In the meantime, combine the half & half, flour, cheese, lemon juice, garlic, salt, and pepper in a mixing bowl; whisk until thoroughly incorporated.
Whisk the prepared sauce into the skillet and bring to a slow boil while still whisking to prevent curdling.
Transfer the chicken back to the skillet and lower the heat to a simmer.
Cook for 3 to 4 minutes or until the asparagus is tender and the chicken is cooked through.
Remove the skillet from the heat. Spoon the sauce over the chicken breasts and top with lemon slices.
Garnish with parsley and cheese, and serve.