For the lobster and basil oil: Combine oil and basil in a blender and blend for 2 minutes. Strain into a bowl and season with salt and pepper.
Heat the grill to high. Thread 2 skewers through the tails lengthwise so that the meat lies flat on the grill. Brush the flesh with basil oil and season with salt and pepper. Grill, flesh side down until slightly charred, about 2 to 3 minutes, flip over, brush top with more of the oil and continue grilling until just cooked through, about 2 minutes longer. Transfer to a serving platter and keep warm.
For the grilled corn-coconut milk sauce: Heat grill to high.
Remove corn from water, place on the grill, close the cover and "steam" in the husk until tender, about 10 minutes. Remove from the grill, let cool slightly and remove the husks. Brush the corn with oil, season with salt and pepper and grill until the kernels are slightly charred. Transfer to a cutting board and let cool slightly. Remove the kernels using a chef's knife and add to a bowl.
Heat about 1 tablespoon of oil in a medium saucepan over high heat. Add the onion and cook until soft. Add the garlic and cook 30 seconds. Add the corn and toss to coat in the mixture. Pour in the coconut milk, bring to a simmer and let cook for 5 minutes. Cool slightly; add to a food processor and puree. Stir in the basil and season with salt and pepper, to taste. Transfer to a serving bowl and serve with the lobster.