Prepare the lobsters for grilling: Set each on its stomach with its head facing inward on a steady, clean cutting board. Holding the lobster down against the board in your nondominant hand, steady the tip of a sharp chef's knife 2 inches in from the tip of the lobster's head. Firmly press the tip of the knife downward through the shell, then swiftly bring the blade down toward the cutting board to cut cleanly through the head. Transfer the lobsters to a large rimmed baking sheet.
Tear off the tails and large claws, and rinse or remove any innards as needed. Make a long vertical cut through the soft underside of the tails, being careful not to cut through the hard outer shell. Set aside the large claws and tails (save the bodies and small claws for another use).
In a medium pot, add the potatoes and enough water just to cover; season generously with salt. Cook over medium-high heat until tender, 10–12 minutes. Drain and keep warm.
Meanwhile, preheat a grill to high heat. Rub the reserved lobster claws and tails, tomatillos, corn, and garlic heads all over with olive oil and season with salt; place on the baking sheet and set next to the grill.
Set the corn and garlic at the edges of the grill where the temperature is slightly cooler. Put the lemons, cut sides down, in a hotter zone. Add the lobster claws and tails to the hottest area of the grill in a single layer. Remove the lemons once well charred on the cut sides, about 5 minutes. Cook the lobster parts, turning as needed, until the shells are bright red and charred in places and the tail meat looks opaque, 6–7 minutes for the claws and 7–9 minutes for the tails. Cook the corn and garlic, turning and stacking the cobs occasionally to avoid overblackening, until the kernels and garlic heads are charred and tender, about 15 minutes. Remove everything to the baking sheet and keep warm. Quickly add the tomatillos to the grill and cook, turning occasionally, until well charred and softened but not mushy, 3–4 minutes. Remove to the baking sheet.
Toss the boiled potatoes with the tomatillos and season the mixture with more salt, pepper, and oil; toss again to coat.
To serve, line a large platter with fresh dill and arrange all ingredients together, including the oysters, or serve the lobster parts with the vegetables and shucked oysters on the side. Season the melted butter generously with salt, and brush some of it on the lobster and corn. Serve hot with more butter on the side for dipping.