Tip:Try tying the fillets with dark green scallion tops instead of using toothpicks. Dip the green tops in boiling water for 15 seconds so they’re pliable and tie in a loose knot around the rolled fish fillets.
Spread the fillets on a work surface, skin side down. Season lightly with salt and white pepper. Divide the 1 Tbs. grated ginger among the fillets and spread as evenly as you can. Sprinkle about 1 Tbs. of the chopped basil over the fillets. Roll each fillet, starting at the thicker end. Secure each roll with a toothpick (or green scallion tops (see the tip below). Sprinkle the rolls lightly with a little more salt and white pepper. Set aside.
In a small (4-quart) Dutch oven with a tight-fitting lid (or other heavy-duty pot just large enough to hold the fish snugly), heat the oil over high heat until it shimmers.
Add the shallot, ginger matchsticks, and garlic. Sauté for 1 min., stirring constantly. Add the celery and sauté for 30 seconds, stirring constantly. Add the chile and continue to sauté, stirring, until the celery starts to soften, about 2 min. Add the chicken broth and heat for 2 minutes. Turn off the heat and arrange the rolled fish in a single layer over the celery mixture. Pour the coconut milk over the fish and turn the heat to high. As soon as the coconut milk comes to a simmer, reduce the heat to medium low and cover. Simmer until the fish is opaque and cooked through, 8 to 10 min. If you’re unsure, flake apart a section to see if it’s done. Remove the pot from the heat.
Carefully transfer the fish with a slotted spoon or fish spatula to a small, warm platter.
Return the pot to high heat. Add the remaining basil, along with the scallions, fish sauce, lime juice, mirin, and lime zest. Bring the broth just to a simmer. Taste the broth and add more salt or mirin if needed.
For each serving, place a small mound of hot jasmine rice in a shallow bowl, top with a fish roll, and remove the toothpick. Ladle the hot broth over each fish roll and serve immediately.