Preheat a grill or grill pan to medium. Whisk 4 tablespoons of the coconut milk, the lime juice, ginger, sugar and 1/4 teaspoon salt in a small bowl.
Tear off a long strip of aluminum foil and place on half of the grill. Arrange the bok choy on a plate and brush with the remaining 3 tablespoons thick coconut milk. Season with salt.
Arrange the bok choy cut-side down on the grill, with the leafy tops resting on the aluminum foil. Cover the grill and cook the bok choy, flipping once, until you get nice grill marks on both sides, 6 to 7 minutes total. Transfer the bok choy to a medium bowl and cover with plastic wrap to steam, 3 to 5 minutes (this will allow the stems to cook through without grilling the leaves to a crisp).
Divide the bok choy among 4 plates and drizzle with the coconut-ginger dressing.
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