Heat a large saucepan over medium-high heat. Add butter and olive oil; heat until melted. Add pork chops; cook until browned, about 2 minutes per side. Add shallot and garlic; cook until tender, about 1 minute. Toss in kale, mushrooms, onion, and red bell pepper; cover and cook until wilted, about 5 minutes. Season with salt, pepper, and pepper flakes.
Pour chicken broth and Dijon mustard into the saucepan. Bring to a high simmer. Reduce heat to medium-low. Stir in arrowroot; cook until broth is reduced by half and sauce has thickened, about 5 minutes. Divide between 2 pasta bowls.