Using a vegetable peeler, peel five 3-inch strips of lemon peel, making sure to avoid white pith; reserve lemon for later. Add lemon peels and 2 tablespoons olive oil to a large pot.
Heat oil over medium and cook, stirring occasionally, until lemon peels start to curl and edges start to toast, about 3 minutes. Add 2 sprigs rosemary and continue to cook until lemon peels are lightly browned and rosemary is fried, 3–4 minutes more (the peels should look on the verge of burning around the edges and you should get a faint whiff of fruitiness).
Remove pot from heat and, using tongs, transfer lemon peels and rosemary to a paper towel-lined plate; set aside to cool. Once cool, finely chop fried peels and rosemary; reserve for garnish.
Season 4 chicken thighs liberally with kosher salt.
Place pot back over medium-high heat and place chicken, skin side down in oil and cook, until golden brown, 6–8 minutes. Flip and continue to cook until browned on second side, about 4 minutes. Transfer chicken to a plate.
Add 10 ounces halved and peeled pearl onions and 1 tablespoon minced fresh rosemary to drippings in now-empty pot, season with ¼ teaspoon kosher salt and toss to coat in drippings. Cook, without moving, until onions start to brown, 3–4 minutes.
Reduce heat to medium and stir in 4 sliced cloves garlic and cook, stirring occasionally until garlic is golden brown and fragrant, about 1 ½ minutes.
Deglaze pot with ¾ cup dry vermouth, scraping up any browned bits from pan, then return chicken back to pot, cover and cook until chicken reaches 172ºF on an instant-read thermometer, 12–15 minutes, flipping chicken over halfway through.