Prepare the duck: Season the duck breasts on both sides with and place in the fridge uncovered for at least 3 hours or up to overnight.
Make the sauce: In a 2-quart saucepan over medium heat, add the olive oil. When the oil is hot, add the onions and carrots. Season with salt and cook, stirring often, until soft, 6 to 8 minutes. Add the thyme, bay leaf and wine, and reduce by half over medium-high heat, 3 to 4 minutes. Add the chicken stock, bring to a boil then reduce over medium heat to 1 cup, 40 to 50 minutes. Strain and reserve the liquid, then transfer it back into the clean saucepan. Bring to a boil and reduce to a ¼ cup, 8 to 10 minutes. Stir in the butter until the sauce is smooth and silky. Keep warm. Make ahead: The sauce can be made the day before, covered and chilled.
Meanwhile, make the wheat berries: In a small saucepan over medium heat, the olive oil. Add the shallots and garlic, season with salt and pepper and cook, stirring often, until softened, 2 to 3 minutes. Add the wheat berries and bay leaf and cover with 1½ cups water and bring to a boil. Reduce the heat, cover the pan and simmer until the wheat berries are al dente, 35 to 40 minutes. Uncover and continue to cook until tender and most of the liquid has evaporated, another 10 to 15 minutes.
Remove the duck breasts from the fridge 30 minutes before cooking. Place the duck breasts skin-side down in a dry, cold skillet and place over medium heat. As the fat begins to render, pour the excess fat from the skillet into a bowl and save for another use. Continue to pour out the fat until the skin becomes crisp and golden brown, 12 to 15 minutes. Flip the breast over and sear for 1 minute. Transfer to a cutting board. Let the duck rest for 5 minutes before carving at an angle against the grain.
Meanwhile, make the endive salad: In a medium bowl, toss the endive, scallion, parsley, orange juice and zest, and vinegar. Season with salt.
Spoon the wheat berries onto two plates. Divide the duck breast slices between the two plates and sprinkle with tōgarashi, then drizzle with sauce and top with the endive salad.