Place a rack in the center of your oven and preheat the oven to 375 degrees F. Season the chicken with salt and pepper.
In a large, ovenproof skillet, melt 2 tablespoons of butter over medium heat. Add the chicken and sear on both sides until deeply golden brown, 10 to 12 minutes total. Transfer to a plate.
Reduce the heat to medium low. Add the remaining tablespoon butter, then the green beans, shallot, garlic, and Italian seasoning. Stir to combine. Cook for 7 minutes, stirring often, until the green beans are crisp-tender. Remove from the heat. Zest the lemon into the pan, then squeeze in the lemon juice. Slowly pour in the coconut milk while stirring to combine. Return the reserved chicken to the skillet.
Place the skillet in the oven and cook for 15 to 25 minutes, or until an instant-read thermometer inserted in the thickest portion of the chicken registers 165 degrees F and the juices run clear (the cook time will vary based on the size of your chicken). Top with parsley. Serve with extra lemon wedges and rice, pasta, and/or fresh bread for mopping up the sauce, if desired.