Set up a water bath with an immersion circulator, adjusting it to 185°F (85°C).
Slice the carrots in half lengthwise, then cut them diagonally into pieces about 1/2-inch thick.
In a resealable freezer bag or vacuum seal bag, combine the carrots with 2 tablespoons of vegetable oil, beef broth, honey, lemon juice, thyme sprigs, and cayenne pepper. Seal the bag, ensuring to remove as much air as you can before placing it inside another freezer bag, removing all the air, and sealing again.
Immerse the bag in the water bath until fully submerged, securing it to the side of the container. Use a weight if necessary to keep the bag from floating. Cover the container with aluminum foil to conserve heat and limit evaporation, and cook for about 70 minutes, or until the carrots become tender.
Pour the contents of the bag through a fine mesh strainer over a medium skillet; set the carrots aside in a separate bowl.
Put the skillet over medium heat and let the liquid come to a simmer. Let it cook down to roughly 3 tablespoons, which should take 3 to 5 minutes. Take the skillet off the heat and incorporate the remaining 2 tablespoons of vegetable oil, stirring until the sauce becomes smooth and glossy.
Return the carrots to the skillet, gently mixing them with the sauce, and finish with a light sprinkle of sea salt flakes. Serve while warm.