Place a rack in the middle of the oven; preheat to 300°. This is an important temperature for this recipe, so if you don't have an oven thermometer already, now is the time to get one! Pat chicken dry with paper towels. Place on a small rimmed baking sheet. Season the whole chicken all over with 1 tablespoon of Diamond Crystal or 1 teaspoon of Morton kosher salt and lots of freshly ground black pepper, making sure to season the inside cavity.
Whisk 5 tablespoons of gochujang and ¼ cup of extra virgin olive oil in a medium bowl until combined. Finely grate 3 garlic cloves (from one of the heads of garlic) into gochujang oil. Peel the fresh ginger (a spoon can get the job done), then grate into gochujang oil; whisk to combine.
Cut what's left of the head of garlic in half crosswise. Repeat with second head of garlic. Stuff
2 garlic halves inside cavity of chicken. Tie legs together with kitchen twine.
Using a pastry brush, brush half of the gochujang oil over the chicken.
Toss those baby Yukon Gold potatoes and the remaining 2 garlic halves and 2 tablespoons of extra virgin olive oil in the remaining gochujang oil until well-coated. Season lightly with salt and pepper and toss again to combine.
Arrange the potatoes in a 12" cast-iron skillet, scooting them toward the edges of the pan to make space for the chicken. Nestle garlic halves (cut sides down) at the center of the skillet. Place the chicken over the garlic as it roasts, it will infuse the fat (and thus, the potatoes) with flavor. If any potatoes have shimmied their way under the chicken, use tongs to arrange them around it (they won't cook at the same rate if they're under the chicken).
Roast the chicken and the potatoes, turning the potatoes once or twice to coat in the juices and oil that accumulate in pan, until the potatoes are very tender when pressed with the back of a spoon and the chicken skin is deep reddish-golden brown in color, about 2½ to 3 hours. When you wiggle the legs of the chicken, they should feel loose in the joints, meaning the meat is fall-apart tender. Transfer the chicken to a cutting board and let rest for about 10 to 15 minutes. Don't skip this step: Letting the chicken rests helps the juices in the meat to settle (in other words, it makes the meat juicier).
Meanwhile, use the back of a large spoon or a potato masher to gently smash those potatoes in the skillet, exposing some of their flesh to juices underneath to soak them up.
Finish the potatoes: Thinly slice the scallions in long diagonal. Cut the limes in half. Cut one half into wedges and set aside. Stir 2 teaspoons of honey and the juice of the remaining half lime into the potatoes. Taste the potatoes and season with more salt if needed. Top with sliced scallions.
Carve the chicken, then arrange its pieces over the potatoes and scallions. Serve right out of the skillet with the remaining lime wedges alongside for squeezing, and squeeze out the sweet, slow-roasted garlic cloves as you wish.