Trim the stem end and the dark green tops from the fennel; reserve a few fronds for garnish. Cut fennel bulb lengthwise into quarters; cut across into 1 inch pieces. Chill in a bowl of iced water 20 minutes; drain.
Place the figs in a small bowl and cover with boiling water; let stand 10 minutes; drain and cut in half.
Sprinkle the chicken with salt and pepper. Heat the butter and olive oil in a large deep skillet until sizzling. Add the chicken, a few pieces at a time and cook until lightly browned, about 5 minutes per side.
Remove the chicken to a side dish as it browns. Leave a thin film of oil in the bottom of the pan: discard remaining oil.
Add the fennel, leeks, and carrots to the pan. Cook, stirring, until the vegetables are softened, about 5 minutes.
Add the figs and fennel seeds to the pan. Cook, stirring, until the cut sides of the fig are lightly browned, about 5 minutes.
Add the garlic; cook, stirring, 1 minute.
Add the chicken broth and the wine. Heat to a boil; boil, uncovered, over high heat until reduced by half, about 5 minutes.
Add the chicken and the juices to the pan. Cover and cook over medium low heat until the chicken juices run clear when pierced with a fork, about 15 minutes.
With a slotted spoon transfer the chicken and vegetables to a warmed serving platter; cover to keep warm.
Boil the juices over high heat until reduced and slightly thickened, 1 to 2 minutes.
Pour the juices over the chicken; garnish with fennel fronds and serve.
Makes 4 servings.