Chicken leg lollipops with sweet and sour glaze, soft polenta and chicken-roasted carrots

Grab each drumstick by the meaty end; about a third of the way up from the ankle, cut down to the shin bone and saw through the meat and tendons while rotating the leg until the bone is circled.

Scrape the meat off the bone toward the ankle, push it down around the ankle joint, set the belly of the knife just under the joint and strike the spine of the knife to cut clean through the bone; remove any bone shards. Trimmings go in the skillet for stock.

Season the legs well on every surface with salt and pepper, toss in a light dusting of cornstarch, then toss with a little oil.

Roast at 400F/200C, convection if available, for about an hour to start.

Brown the bone and skin trimmings in a little oil over medium heat until golden.

Cut thin angled slices from the upper middle of the green onions and reserve for garnish; roughly slice the tops and bottoms for the stock. Smash and peel the garlic.

Add the onion pieces and garlic to the trimmings and stir until everything has some browning, not too much.

Push aside, add the tomato paste and brown it. Add half a bottle of white wine and scrape the bottom of the pan.

Add the dried mushrooms and enough water to submerge, a little salt and pepper; bring to a boil and reduce to a spirited simmer, just under an hour, topping off with water once or twice.

Whisk the cornmeal into about a quart of salted boiling water. Turn the heat all the way down, cover and leave at least half an hour.

Roast the small carrots whole; halve long ones, and halve and split the thick tops of long thick ones.

After an hour, release the chicken from the tray, push it to one end, add the carrots with a drizzle of oil and a pinch of salt, toss, and respread the chicken in a single layer. Roast another half hour.

Strain the stock through a sieve set in a bowl, squeezing the solids with a spoon; discard the solids, rinse the pan, return the stock and bring to a simmer.

Make a smooth slurry of a couple tablespoons of starch and a tiny splash of water, working out lumps with a finger; drizzle it in slowly while stirring briskly, using only as much as needed, and simmer to thicken.

Season the glaze with soy sauce, a splash of vinegar and a lot of honey; it should taste too strong on its own and look slightly too thin. Add more soy sauce for salt.

When the legs have had 90 minutes and the carrots are about two-thirds cooked, swirl each leg in the glaze one at a time, keeping the bone handle clean, for a first coat; keep warm on low.

Finish the polenta with a few tablespoons of butter, salt and pepper to taste, and more water if wanted, remembering it firms up as it cools.

Put the carrot tray on the heat, add a little water, scrape up the brown bits, and cook the carrots to the tenderness wanted while reducing the liquid to a glaze.

Plate the polenta and carrots, give the chicken a final glazing, and sprinkle with the reserved green onions.