Preheat an outdoor grill for medium-high heat and lightly oil the grate. Roll out a little more than 1 foot aluminum foil. Repeat with a second sheet and place on top of first.
Spoon lobster base in the center of the foil; top with butter. Place Dover sole on top of butter and add shallot, red bell pepper, and garlic. Drizzle olive oil over garlic layer and top with parsley, zaatar, salt, and pepper. Gently roll fish to mix ingredients. Wrap foil around mixture and seal tightly.
Place foil packet on the edge of the direct heat zone on the preheated grill; cook until fish flakes easily with a fork, about 20 minutes. Take packet off grill and allow to cook from the steam for 5 minutes more.
