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Bring a large pot of water to a boil over high heat. Cook pasta 3 minutes less than time according to package instructions. Add broccoli and continue cooking for the remaining 3 minutes. Drain and keep warm.
Meanwhile, toss chicken with flour, salt and pepper in a medium bowl. Heat 2 teaspoons oil in a large non-stick skillet over medium-high heat. Add the chicken, leaving excess flour in the bowl. Cook chicken, stirring occasionally, until browned on one or two sides, about 4 minutes. Remove to a plate and keep warm (it will continue cooking in step 4).
Wipe out skillet with a clean paper towel. Reduce heat to medium.
Heat the remaining 2 teaspoons oil in the skillet. Add garlic, and cook stirring until fragrant, 30 to 90 seconds. Stir in the chicken and any juice from the plate.
Dust on the remaining flour from the bowl, and stir to combine. Add broth and stir until the flour is incorporated. Increase heat to high and bring to a simmer stirring often. Stir in milk, half and half, Dijon and thyme, reduce heat to medium and return to a simmer, stirring. Simmer, stirring occasionally until the chicken is cooked through and the sauce is thick, about 4 minutes. Remove from heat and add 1 cup Parmesan. Stir until the sauce is smooth. Add pasta and broccoli, and stir until combined. Serve topped with additional Parmesan.