Preheat grill to high. Brush the cut side of the lobster with oil and season with salt and pepper. Place the lobster on the grill, shell side down, and grill until lightly golden brown and just cooked through. Remove the meat from the lobster and coarsely chop.
Heat a tablespoon of oil in a small saute pan on the grates of the grill. Add the garlic and cook until lightly toasted. Place 4 tortillas on a work surface and divide the half of the cheese, lobster, and garlic among the tortillas. Season with salt and pepper, to taste.
Place another tortilla on top and then divide the remaining cheese, lobster, and garlic over top. Place the remaining tortillas on top to make 2-layer quesadillas. Brush tops of the tortillas with oil and then place the quesadillas on the grill, oiled side down. Grill until golden brown, about 2 to 3 minutes. Carefully flip the quesadillas over, close the cover of the grill, and grill until the cheese has melted, about 2 to 3 minutes longer. Remove from the grill, cut each quesadilla into quarters, and serve with the various toppings.
Yellow Tomato-Tarragon Relish:
Combine all ingredients in a medium bowl. Season with salt and pepper, to taste.
Roasted Red Pepper and Ancho Salsa:
Place ancho chiles in a medium bowl, cover with boiling water, and let sit at room temperature for 1 hour. Remove from soaking liquid, coarsely chop, and place in a blender with the garlic, 1/4 to 1/2 cup of the soaking liquid, 2 to 3 tablespoons canola oil, and the pine nuts; blend until smooth. Pour the mixture into a medium bowl.
Preheat grill to high. Brush peppers with oil and season with salt and pepper. Place on the grill and grill until charred on all sides. Remove, place in a bowl, cover with plastic wrap, and let steam 15 minutes. Remove skin and cut into julienne strips. Transfer the peppers to the bowl with the ancho puree and add the vinegar. Chop the cilantro, add it to the salsa, and season with salt and pepper, to taste.
Grilled Serrano and Avocado Salsa:
Preheat grill to high. Brush serranos and green onions with oil and season with salt and pepper. Grill until charred on all sides. Remove from the grill and finely chop.
Place avocado in a medium bowl. Add the grilled serranos, green onions, lime juice, and 3 tablespoons oil and stir gently until combined. Season with salt and pepper, to taste.