For the sauce:
1/3 cup unsweetened coconut milk
2 tablespoons smooth or chunky all-natural peanut butter
1 tablespoon soy sauce or tamari
2 teaspoons Asian chile-garlic paste or sambal oelek
3/4 teaspoon packed light brown sugar
For the stir-fry:
1 medium head Napa or Savoy cabbage (about 2 pounds)
2 medium carrots
2 tablespoons vegetable oil
Salt
1 teaspoon freshly squeezed lime juice
2 tablespoons coarsely chopped roasted peanuts (optional)
Steamed rice or cooked rice noodles, for serving
For the sauce: Whisk all of the ingredients together in a medium bowl to combine.
For the stir-fry: Quarter the cabbage lengthwise, then cut the core out of each piece. Cut the cabbage crosswise into 1-inch-wide pieces. Wash and thoroughly dry the cabbage pieces. Peel the carrots and cut crosswise into 1/4-inch-thick rounds.
Heat the oil in a wok or large frying pan over medium-high heat until shimmering. Add the carrots and half of the cabbage and cook, stirring occasionally, until starting to wilt. Continue adding the cabbage and cooking it down until all of it is added. Season with salt and cook until all of the cabbage is starting to wilt and exudes water.
Reduce the heat to medium, add the sauce, and continue cooking, stirring occasionally, until the sauce thickens slightly (it will thin out when mixed with the cabbage liquid, then thicken to the texture of whole milk) and the cabbage is crisp-tender, about 3 minutes. Remove from the heat, add the lime juice, and stir to combine. Top with the roasted peanuts, if using, and serve immediately over steamed rice or cooked rice noodles.