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Arrange a rack in the middle of the oven and heat the oven to 450ºF. Meanwhile, prepare the cabbage and chicken.
Cut 1/2 large or 1 small cabbage into at least 1-inch thick slabs. Arrange cut-side down in a single layer in a 10 or 12-inch cast-iron or other oven-safe skillet or frying pan, trimming or cutting them as needed to fit in a single, tight layer. Season with 1 teaspoon of the kosher salt and a few grinds of black pepper.
Mix 1/2 teaspoon dried thyme, the remaining 1 teaspoon kosher salt, and 1/4 teaspoon black pepper together in a small bowl. Cut 3 tablespoons unsalted butter into small pieces. Scatter about 1/3 of the butter pieces over the cabbage.
Remove and discard any giblets inside a whole chicken, then pat the chicken dry with paper towels. Gently run your fingers between the skin and meat over the breasts and legs to loosen and separate without tearing. Tuck the remaining pieces of butter under the skin. Season the outside of the chicken all over with the thyme mixture. Place the chicken breast-side up on the cabbage, then tuck the wing tips under the body.
Roast for 30 minutes. Meanwhile, peel 2 medium apples. Quarter and core the apples for 8 total pieces.
Remove the pan from the oven. Tuck the apple pieces around and a little bit under the chicken. Roast until the chicken is golden brown and an instant-read thermometer inserted into the thickest part of the thigh registers 165ºF, 30 to 40 minutes more.
Remove the pan from the oven. Tilt the chicken slightly (I like to insert the handle of a wooden spoon into the neck cavity to tilt it) so the juices in the cavity run out into the pan. Transfer the chicken to a clean cutting board and let rest at least 15 minutes. Meanwhile, nestle the apple pieces into the cabbage. Return the pan to the oven and roast until the cabbage is browned, about 15 minutes more.
Carve the chicken and serve with the cabbage and apples.