Heat vinegar and cider in saucepan over high heat. When simmering, add salt and honey and stir until dissolved. Remove from heat, and stir in ice until dissolved and mixture is cooled.
Put the brine and the chicken in a large pot or bowl, and add enough additional ice and cold water to just cover top of the chicken. Soak, refrigerated, for a few hours or overnight.
Rinse off your chicken and dry thoroughly. Set on a plate to dry more and let the chicken come to room temp while you prep your marinade and vegetables.
Preheat oven to 425 degrees. Trim carrot tops down to 1/4 inch, then scrub the roots (or peel). Reserve feathery greens, removing enough to chop for 1 minced tablespoon.
Mix 1 tablespoon minced carrot greens with 5 tbs. of the butter, mustard, sage and garlic. Set aside.
Toss carrots, Brussels sprouts and shallots with one tablespoon of your compound butter and a little bit of salt and pepper. Reserve all trimmings in a freezer bag for stock later.
In a roasting pan with rack, rub the bottom of the pan with remaining 1 tbs. butter, and put in the veggies. Place rack over top.
Trim off any excess fat from your chicken. Use your index finger to create a cavity between the breast and the skin on each side, being careful not to tear the skin. Push some of your compound butter into each side, covering as much of the breast as you can. You can also work from the cavity to separate the skin from the thighs and stuff with butter mix. Rub any remaining compound butter over the outside of the chicken. Salt and pepper the cavity, then stuff with the sage and reserved carrot greens. Sprinkle the chicken lightly all over with salt and pepper. Set on rack breast side up and place in the oven with the cavity facing the back wall.
Roast for 1 hour or more, until thermometer inserted into thigh meat reads 165. Move chicken on rack to a plate, and let rest 10 minutes or more. Transfer roasted vegetables to a bowl, taste and season as needed, and add a splash of apple cider vinegar to brighten. Cover.
While your chicken cools, pour off all but a tablespoon of fat from the pan (keep all the good brown bits in there). Warm over medium high heat, then add vermouth and cider. Cook, scraping as you go, until the liquid is very reduced but not totally evaporated, about 5 minutes. Add in stock, cook until reduced by about half, another 5 minutes or so. Reduce heat to medium-low and add in the butter. Cook, stirring, until the butter is melted and the sauce is satiny.
Carve bird and serve with pan sauce, carrots and sprouts.
1. Keep chicken carcass in freezer, along with your vegetable trimmings, and add to this whenever you can. When you have 1 to 2 cups of vegetable trimmings, place those and the frozen chicken carcass in a slow cooker, add some smashed peppercorns and a bay leaf, and any sadder looking vegetables from your fridge and cover it all with water. Add in a splash of vinegar (I've read this helps break down the bones, and if it doesn't it tastes good anyway) and set to cook on low for 10 hours. Allow to cool a little, strain, and then refrigerate or freeze.
2. If you don't have a roasting pan don't fret; I use an oven-safe skillet and put the rack from my toaster oven over top, it works just fine. And if you don't have anything that would serve as a rack, you can sit the bird right on top of the veggies; the bottom of the chicken and the covered vegetables won't get browned but it will still be tasty.