How to Make It
In a shallow dish, combine 2 tablespoons of the olive oil with 1 tablespoon of the thyme and the garlic and season with salt and pepper. Add the apricot halves and toss to coat. Let stand for 1 hour.
Meanwhile, preheat the oven to 400°. In a medium saucepan, combine the apple juice, vinegar and sugar and boil over moderately high heat until slightly thickened, about 15 minutes. Add the cherries and boil until tender, about 3 minutes. With a slotted spoon, transfer the cherries to a bowl. Add the shallots to the saucepan and boil over high heat until the liquid has reduced to 1/2 cup, about 4 minutes. Return the cherries to the saucepan. Stir in the remaining 1 teaspoon of thyme and season with salt and pepper. Keep the cherry sauce warm.
Light a grill. In a medium bowl, mix the remaining 1 1/2 tablespoons of olive oil with the lime juice and season with salt and pepper. Add the fennel and toss well.
Grill the apricots over a medium-hot fire until lightly charred and tender, about 3 minutes per side. Transfer to a plate and keep warm.
Heat a large ovenproof skillet. Season the duck breasts generously with salt and pepper. Add to the skillet, skin side down, and cook over moderately high heat until some of the fat has been rendered, about 3 minutes. Transfer the skillet to the oven and roast the duck breasts for 3 minutes. Turn the breasts and roast for about 3 minutes longer, until medium-rare. Transfer the breasts to a carving board to rest for 5 minutes.
Toss the arugula and chives with the fennel salad and arrange on 4 plates. Slice the duck breasts crosswise 1/4 inch thick; arrange the slices on the plates and top with the cherry sauce. Garnish with the grilled apricots and serve.