Pat duck legs dry. Heat the duck fat in a Dutch oven over medium heat.

Place duck legs skin side down. Render and crisp until deeply golden, about 8 minutes. Remove and set aside.

In rendered duck fat, place quartered onion cut side down. Press gently and sear until charred, about 5 minutes. You'll know it's ready to flip once it releases from the pan. It sticks because the crust is still forming, so don't nudge it around if it's not releasing. Remove and reserve.

Pour off excess fat, leaving 2 tbsp in the pot.

Add carrots and celery, season lightly with salt, and cook until softened, 6 minutes.

Deglaze with bourbon, scraping up browned bits. Reduce slightly, 2 minutes. Taste it as you go. If you feel astringency at the back of your throat, the alcohol hasn't cooked off yet. Keep going until it turns round and caramel.

Add red wine and simmer until reduced by half, about 8 minutes. Add chicken stock and bring to a simmer.

Return duck legs, skin side up, along with the charred onion.

Add the 8 figs, thyme, bay leaves, and pee-wee potatoes. Liquid should almost cover the duck.

Bring to a simmer. Cover and transfer to oven at 325°F. Cook for 1 hour 45 minutes, until duck is tender.

After 30 minutes, begin checking potatoes. Once fork-tender, remove with a slotted spoon.

Remove duck to a platter. Lift the figs out with a slotted spoon and set them aside. Discard onion, thyme, and bay leaves.

Set a fine mesh strainer over a clean saucepan. Press the braised figs through it with the back of a wooden spoon, then strain the braising liquid through after them. The pulp goes into the sauce and the skins and seeds stay behind. Don't press what's left in the strainer or the sauce goes cloudy.

Add the last 2 figs, cut side down.

Skim fat from surface. Reduce until glossy, about 10 minutes. If you're reducing for a sauce, you're looking for nappe. It coats the back of a spoon, and when you draw your finger through it, it leaves a distinct line.

Lift the last 2 figs out and press them through a clean strainer back into the sauce, the same as the first 8.

Place duck legs in a skillet with the potatoes. Brush skin with reduced sauce.

Roast at 400°F for 5 to 7 minutes until the skin begins to crisp.

Remove, brush with more sauce, and return to oven. Repeat once or twice for a lacquered glaze.

Warm the sauce back through until it steams. Off the heat, whisk in cold butter until emulsified and silky. Butter breaks in a 400°F oven, so the skin gets the plain reduction and this sauce goes on the plate.

Pile the arugula onto the plates and set the duck and potatoes over it. The leaves wilt where they touch the hot skin and take on the sauce, which is why it belongs under the duck rather than beside it. Spoon the sauce around the duck. Scatter the chopped thyme over the duck.