Preheat oven to 400° with rack in middle position. In a large bowl, dissolve 1/4 cup salt in 2 qts. cold water; add chicken. Let sit at room temperature 45 minutes.
Meanwhile, bring a small pot of water to boil. Separate garlic into cloves, put in a metal bowl, and fit another bowl over garlic so it forms a domed lid. Vigorously shake bowls together for several seconds, as though you're mixing a cocktail; the cloves should shed their skins. Repeat if necessary.
Boil peeled garlic 30 seconds. Drain, rinse with cold water, trim any brown spots and hard stem ends, and set aside.
Heat 3 tbsp. oil in a large pot over medium heat. Meanwhile, tear chard and beet greens into large pieces. When oil is just hot, add 1/3 of greens and let wilt down, stirring occasionally. Add remaining greens the same way. Stir in chiles, zest, and salt to taste and cook, covered, stirring occasionally, until greens are tender, 10 to 15 minutes. Set aside, covered.
Remove chicken from brine, dry thoroughly on kitchen towels, and season lightly with salt. Heat 2 tbsp. oil in a 12-in. ovenproof frying pan over medium-high heat. Working in 2 batches, brown chicken on both sides, about 4 minutes per side, and transfer to a baking pan in a single layer.
Add wine, chicken broth, garlic, and herbs to pan and bring to a simmer, scraping up browned bits. Return chicken to pan skin side up and roast in oven, uncovered, until an instant-read thermometer inserted in thickest part of thigh reads 160°, 12 to 14 minutes. Broil chicken 5 minutes to re-crisp skin.
Meanwhile, stir lemon juice into greens and reheat, uncovered to evaporate some liquid, until hot, about 5 minutes. Stir in remaining 1 tbsp. oil. Spoon greens onto a platter and top with chicken and garlic (remove bay leaves ).
Blend butter and flour into a paste with a fork, then whisk into juices in pan. Boil until slightly thickened, about 3 minutes, and pour into a serving pitcher. Add a few thyme sprigs to the platter if you like and serve with sauce.