Preheat your oven to 170 degrees F or set it to the keep-warm setting.

Pat the chicken completely dry with paper towels. Season both sides generously with salt and pepper. Dredge each piece in flour until lightly coated, then shake off any excess.

Heat 2 tablespoons of olive oil in a large skillet over medium-high heat until shimmering. Add the chicken in a single layer without crowding. Work in batches if needed. Cook 4 minutes on the first side without moving, then flip and cook another 2 to 4 minutes until cooked through and golden. Add more oil before flipping if the pan looks dry.

Transfer the cooked chicken to a sheet pan and keep warm in the oven.

Bring a medium pot of heavily salted water to a rolling boil over high heat. Trim 1.5 inches from the broccolini stems, remove any leaves, and slice larger pieces lengthwise so all pieces are a similar size.

Add all the broccolini to the boiling water at once, cover, and set a timer for exactly 2 minutes. Do not overcook.

Transfer the broccolini to the hot skillet with 1 tablespoon of olive oil over medium-high heat. Saute for 1 minute, then move to the oven with the chicken. If done before the chicken, drop broccolini into cold water immediately to stop the cooking, then saute as directed when the pan is free.

Add the diced shallots to the same skillet over medium-high heat. Add a small drizzle of oil if the pan is dry. Cook for 2 to 3 minutes, stirring occasionally, until soft and translucent.

Pour in the chicken broth and add the thyme sprigs. Bring to a boil, then reduce to a steady simmer for 3 minutes.

Stir in the heavy cream and mustard. Simmer for 5 to 6 minutes, stirring occasionally, until the sauce coats the back of a spoon. Remove and discard the thyme stems.

Return the chicken and broccolini to the pan, or plate and spoon the sauce generously over the top. Serve immediately.