Place a rack in middle of oven; preheat to 375°F. Season chicken legs all over with 3 tsp. salt.

Swirl 1 Tbsp. oil in a large Dutch oven or other heavy pot to coat and heat over medium-high. Place 2 chicken legs, skin side down, in pot, making sure skin is in direct contact with pan. Cook, undisturbed, until dark golden, about 5 minutes. Turn over and cook until golden on other side, about 2 minutes. Transfer to a large plate. Repeat with remaining 2 chicken legs; reserve pot.

Meanwhile, working one at a time, stand a bulb, root end down, on a cutting board and slice off a short end to create a flat surface. Turn cut side down and slice ½" thick, starting from the top end of bulb so the root end can hold the layers in place while you slice. Discard last ¼" (root end) of each bulb and pop out and discard tougher center from slices. Transfer fennel to a large bowl and add half of the grapes, ½ tsp. salt, and remaining 1 Tbsp. oil; toss to coat. Set aside.

Reduce heat to medium and combine broth, honey, chile paste, and remaining 1 tsp. salt in reserved pot. Stir to combine and dissolve salt. Place remaining grapes in pot and arrange chicken legs, skin side up and overlapping if needed, over grapes. Top with dressed fennel and grapes. (Don’t worry if it looks crowded: Fennel will soften as it cooks.)

Transfer pot, uncovered, to oven and roast until fennel is slightly softened, with some crispy, caramelized edges, grapes on top are swollen and browned in spots, and chicken is cooked through, 30–40 minutes. Let cool 5 minutes.

Using a slotted spoon or tongs, transfer grapes and fennel to a bowl and chicken to a platter. Place pot over medium-high heat and bring braising liquid to a simmer. Cook until reduced by half (you should have about 1 cup liquid), 8–10 minutes. Remove from heat and stir in vinegar and fennel fronds.

Spoon sauce over chicken and serve with fennel and grapes alongside.