Preheat grill to medium high heat.
Slice bok choy in half lengthwise.
Melt 1 tablespoon of butter in the microwave. Brush both sides of the bok choy with the melted butter. Season with salt and pepper.
Grill the bok choy, cut side down, for 3–4 minutes, or until charred in spots. Flip and cook for another 2–3 minutes, or until tender. Remove from the grill and place on a serving platter.
Meanwhile, melt the remaining 2 tablespoons of butter in a sauté pan over medium heat. Continue to cook, picking the pan up to swirl occasionally, or until the butter is light brown and smells toasted, about 4–5 minutes. If the butter turns dark brown or black, it is overcooked and you’ll need to start again, so watch closely!
Remove from the heat. Stir in the grated ginger and season with a pinch of salt.
Pour the ginger brown butter over the bok choy and serve immediately.