Sprinkle both sides of chicken thighs with salt and pepper. Heat oil in a Dutch oven over medium-high heat until shimmering; add chicken thighs skin-side down and cook without moving them until skin is crisped and well browned, 10 to 12 minutes. Using tongs, turn chicken and brown on second side, about 5 minutes longer. Transfer thighs to large plate.
Pour off all but 1 teaspoon fat from pot. Add the onion and tomato paste and cook over medium heat, stirring occasionally and scraping bottom of pot with wooden spoon, until tomato paste begins to darken, about 3-4 minutes (if bottom of pot becomes very dark and sticky, stir in 1 to 2 tablespoons water). Add garlic and anchovy and cook, stirring constantly, until fragrant, about 1 minute. Stir in tomatoes, chicken broth, and wine, scraping up browned bits with wooden spoon. Add red pepper flakes, thyme, and bay. Remove and discard skin from chicken thighs, then submerge chicken bone-side up in liquid, adding any chicken juices accumulated on plate. Increase heat to high, bring to simmer, cover, then place pot in oven. Cook until chicken offers no resistance when poked with tip of paring knife but meat still clings to bone, 40 to 55 minutes.
While chicken cooks, trim stems from Swiss chard. Roughly chop the leaves.
Also while chicken cooks, simmer balsamic vinegar in a small saucepan over medium-high heat until thick, syrupy, and reduced to 1/3 cup, 3 to 5 minutes (begin measuring volume when vinegar begins to cling to sides of saucepan). Set aside.
Using a slotted spoon, transfer chicken to plate and tent with foil; discard bay leaf. Bring liquid in Dutch oven to simmer over medium-high heat. Add chard greens and cook until wilted, about 2 minutes. Stir about 1/3 cup sauce into balsamic reduction to loosen, then stir the mixture into the sauce. Taste and adjust seasoning with salt and pepper.
Return the chicken and any accumulated juices to sauce, cook until heated through, about 2 minutes, turning chicken once or twice. Use a slotted spoon to transfer chard to serving dish or individual bowls; place chicken thighs on chard, then spoon sauce over. Serve immediately.