Preheat the oven to 400°F.
On a baking sheet, coat the broccoli with the olive oil and a pinch of salt. Roast for 12 minutes, or until fork-tender and crisp.
Bring a large pot of salted water to a boil. Add the pasta and cook according to package instructions. Drain, return to the pot, and toss with olive oil.
Dredge the chicken in flour, shaking off any excess.
Melt 2 tablespoons of butter in a large skillet over medium heat. When hot, add 2 chicken breasts and cook for 3 minutes. Flip and cook for 3 more minutes, until cooked through. Transfer to a plate and repeat with the remaining chicken, adding more butter if necessary.
When all of the chicken is browned, add the lemon juice, chicken stock and capers to the empty skillet. Bring to a boil, scraping up any browned bits on the bottom of the pan. Return the chicken to the skillet and cook for 5 minutes. Remove the chicken to a serving plate.
Over very low heat, whisk the remaining 2 tablespoons of butter into the sauce in the skillet until emulsified. Adjust the seasoning with salt and pepper.
Pour the sauce over the chicken and garnish with parsley. Serve over fettuccine with roasted broccolini on the side.