Prep the Chicken: Slice each chicken breast in half lengthwise, then pound them out thinly. Season both sides with salt and pepper.
Dredge & Sear: Coat each piece lightly in flour, shaking off the excess. Heat oil in a pan over medium-high heat, then cook the chicken until golden brown on both sides. Remove and set aside.
Make the Sauce: In the same pan, sauté the shallot and garlic until fragrant. Pour in the white wine, scraping up any crispy bits (aka flavor). Let it reduce, then stir in the lemon juice, and butter until the sauce is glossy and dreamy. Add in the broccoli, cover and allow to steam in the sauce with the lid on for a few minutes, until the broccoli is fork tender.
Put It All Together: Add in heavy cream. If the sauce has reduced too much add a splash of chicken broth. Return the chicken to the pan, letting it soak up all that goodness. Sprinkle with fresh parsley and serve over rice.