Place a rack in middle of oven; preheat to 300°.

Pat chicken dry with paper towels. Place on a small rimmed baking sheet. Season chicken all over with salt and lots of freshly ground black pepper, making sure to season the inside cavity.

Whisk gochujang and olive oil in a medium bowl until combined. Finely grate 3 garlic cloves into gochujang oil. Peel ginger, then grate into gochujang oil; whisk to combine.

Cut what’s left of the head of garlic in half crosswise. Stuff 2 garlic halves inside cavity of chicken. Tie legs together with kitchen twine.

Using a pastry brush, brush half of gochujang oil over chicken.

Toss baby Yukon Gold potatoes and remaining 2 garlic halves and olive oil in remaining gochujang oil until well coated. Season lightly with salt and pepper and toss again to combine.

Arrange potatoes in a cast-iron skillet, scooting them toward edges of pan to make space for chicken. Nestle garlic halves in center of skillet. Place chicken over garlic.

Roast chicken and potatoes, turning potatoes once or twice, until potatoes are very tender and chicken skin is deep reddish-golden brown, 2½–3 hours.

Transfer chicken to a cutting board and let rest 10–15 minutes.

Use the back of a large spoon or a potato masher to gently smash potatoes in skillet.

Thinly slice scallions. Cut limes in half. Stir honey and juice of remaining lime half into potatoes. Taste potatoes and season with more salt if needed. Scatter sliced scallions over potatoes.

Carve chicken, then arrange pieces over potatoes and scallions. Serve right out of skillet with remaining lime wedges.