Preheat a very clean and very well-oiled grill to high heat.

Lay fish alongside each other on a clean kitchen towel and gently roll into a cylinder to remove excess moisture.  (It is very important that the fish is sufficiently dry.)  Transfer fish to a rimmed baking sheet.  Stuff the cavity of each fish with 6 thyme sprigs and season the insides with 1 teaspoon of sea salt and ½ teaspoon of the black pepper. Pour 1/2 cup of the olive oil over the fish and turn them in the oil to generously coat both sides. Sprinkle the fish evenly on both sides with the remaining 1 teaspoon sea salt and ½ teaspoon black pepper. 

Oil the grill grates one more time, and place the fish directly on the grill. Reduce the heat to medium and cook, undisturbed and turning only once, until the flesh flakes easily and is just cooked through, about 5 minutes per side. (If you try to turn the fish too soon, they will stick to the grill. Be patient here and the fish will easily turn once the skin has crisped.)

While the fish is cooking, place the lemon slices on the grill and brush lightly with some of the remaining 1/4 cup olive oil. Cook until softened and nicely marked on both sides, 2 to 3 minutes per side.

Transfer the fish to a platter and drizzle them with the lemon juice and extra-virgin olive oil.  Sprinkle with the thyme leaves and additional sea salt if desired.  Garnish with the grilled lemon slices.  Serve immediately.