Preheat the broiler. Toss the shrimp, snap peas, garlic, thyme and olive oil in a large bowl; set aside.
Put the potatoes in a pot and cover with water by 1 inch; bring to a boil. Reduce to a simmer and cook until fork-tender, about 10 minutes. Reserve 1/4 cup cooking water, then drain the potatoes and return to the pot. Add the buttermilk, 2 tablespoons butter, 1/2 teaspoon salt and a few grinds of pepper. Mash, gradually adding the reserved cooking water as needed. Stir in 1 tablespoon chives.
While the potatoes are cooking, spread the shrimp and snap peas on a baking sheet. Broil until just cooked through and charred in spots, 6 to 8 minutes.
Dot the shrimp and snap peas with the remaining 2 tablespoons butter and sprinkle with the lemon zest. Toss until the butter melts and the mixture is well coated; season with salt and pepper. Divide the potatoes, shrimp, snap peas and any juices from the baking sheet among plates. Sprinkle with the remaining 1 tablespoon chives.