In a skillet over medium heat, warm the olive oil. Add the shallots and cook, stirring occasionally, until golden, 10 minutes. Add the wine, chicken stock, figs, lemon zest, sugar, and cloves. Decrease the heat to low and simmer until the figs are very soft but still retain their shape, about 30 to 45 minutes. Then increase the heat to high and simmer until the sauce thickens, 5 to 10 minutes. Season to taste with salt and pepper. Set aside.
Preheat the oven to 400 degrees F.
Place the duck breasts on a work surface, skin side up. Using a sharp knife, score the skin in a grid pattern, going through half the thickness of the skin but not through to the meat.
Heat a large ovenproof skillet over medium-high heat. Add the duck breasts, skin side down, and cook until the skin is golden and the fat from the skin has rendered, 6 to 8 minutes. Turn the duck breasts over, season with salt and pepper, and trasnfer the skillet to the oven. Cook until the duck breasts are medium-rare, 6 to 8 minutes.
Remove the skillet from the oven, remove the duck, cover it with foil, and let rest for 10 minutes. In the meantime, warm the fig sauce.
Cut each duck breast on the diagonal into 1/2-inch-thick slices. Fan the slices out on each plate, spoon the warm sauce over the top, and serve.
Cook's note: Can't find duck breast? Try the fig and shallot compote with chicken or pork.