Bring water to a simmer over medium heat, then remove from heat. Add lemongrass ginger tea bags and steep for 10 minutes. Refrigerate until chilled.
In a bowl, combine cooked lobster meat, celery, mayonnaise, lemon juice, chives, dill, salt, and black pepper. Mix gently until evenly coated.
Preheat a skillet or griddle over medium heat. Add butter and melt.
Place hot dog buns in the skillet and toast until golden brown on both sides, about 2 minutes per side.
Fill toasted buns with the lobster mixture.
Serve the lobster rolls with cheese slices, apple slices, and ginger lemongrass iced tea.