Fill a large pot with water and place a sous vide cooker into the water. Set temperature to 140°F (60°C), according to manufacturer's directions.

Pound chicken breasts with a meat mallet to an even 3/4- to 1-inch thickness. Season both sides with Cajun seasoning. Place chicken into vacuum bags and seal using a vacuum sealer.

Lower the bags into preheated water, making sure they are fully submerged; set the timer for 1 hour. When the timer is up, remove chicken from water and immediately place into an ice bath for 5 minutes. Remove chicken from the bags and pat completely dry with paper towels.

Heat butter and avocado oil in a large skillet over high heat until just smoking. Sear chicken in the hot butter-oil mixture for 30 to 45 seconds per side, moving chicken around in the skillet to get all sides evenly browned. Remove to a plate and cover with foil to keep warm.

Make sauce: Melt butter in the same skillet over medium heat. Add garlic and sauté in hot butter until fragrant, 15 to 30 seconds.

Whisk in chicken broth and bring to a simmer. Cook until slightly reduced, 3 to 4 minutes.

Whisk in heavy cream and black pepper; bring to a light simmer. Cook, whisking continually, until thickened.

Stir in Parmesan cheese, 1 tablespoon at a time, until melted and combined. Sprinkle in thyme. Taste the sauce and adjust salt, if needed.