Step 1: Position a rack in the lower third of your oven and preheat to 425 degrees F.
Step 2: Cut each chicken breast in half crosswise n the diagonal into 2 roughly equal portions and sprinkle with 1/4 t each of salt and pepper. Place flour in a shallow dish and dredge chicken lightly; reserve the remaining flour. Heat 2 T. oil in a large skillet over medium heat. Add the chicken, meaty side down; cook until well-browned on the bottom, about 5-6 minutes. Transfer, meaty side up, to a rimmed baking sheet (I lined mine with non-stick foil). Reserve the skillet.
Step 3: Toss the broccolini (or broccoli cuts) with the remaining 2 T. of oil, half the garlic and 1/4 t. salt. Spread out on the other half of the baking sheet.
Step 4: Roast until the broccolini (or broccoli cuts) is tender and an instant-read thermometer inserted into the thickest part of the chicken registers 165 degrees F., 15 to 20 minutes.
Step 5: Meanwhile, whisk 4 t. of the reserved flour and wine in a small bowl until well combined (no lumps). Place the skillet over medium heat, add the remaining garlic and cook, stirring, for 30 seconds. Add broth; bring to a boil of high heat. Add the wine mixture and cook, stirring frequently, until thickened and reduced to about 1 C, 4 to 5 minutes. Add Parmesan, 1 t. thyme and the remaining 1/4 t. each salt and pepper.
Step 6: To serve, divide the chicken, broccolini (or broccoli cuts) and rice among 4 dinner plates. Spoon the sauce over the chicken (and rice and/or broccoli if you like) and sprinkle with the remaining 1 t. thyme.
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