For the chicken: Combine kosher salt, herbs, lemon zest, garlic powder, pepper, coriander, and paprika in a bowl. Rub over chicken, taking care to work it into any folds, crevices, or openings. Cover and refrigerate at least 6 hours and up to 24 hours.

Preheat oven to 425°F. Drizzle chicken with oil, using your hands to ensure it is well coated. Heat a large 12-inch cast-iron skillet over medium-low heat until it begins to smoke, 10 to 12 minutes. Add chicken, skin side down. Reduce heat to low and place the brick lengthwise down the center of the bird. Cook until skin is crisp and golden brown, 12 to 14 minutes.

Remove the brick. With tongs, carefully flip chicken skin-side up. Transfer to the oven and roast until the internal temperature on an instant-read thermometer, inserted between the breast and thigh, reaches 165°F, 15 to 25 minutes. Transfer to a large platter or cutting board and let rest 15 minutes before carving; reserve the skillet.

For the braised fennel: Melt butter in reserved skillet over medium heat. Add fennel, carrots, and leek. Season with kosher salt and pepper. Cook, stirring occasionally, until vegetables are translucent and beginning to brown, 10 to 12 minutes. Add wine or vermouth and cook, stirring, until almost completely evaporated, 2 to 4 minutes. Remove from heat and stir in tarragon. Sprinkle with flaky salt and serve alongside chicken.