Heat a tablespoon of oil in a small saute pan over medium heat. Add the garlic and cook until lightly toasted.
Place 8 of the tortillas on a work surface and divide the cheese, lobster and garlic among the tortillas. Season with salt and pepper. Place the remaining tortillas on top of the filling. Brush the tops of the tortillas with oil, and then place the quesadillas on the grill, oiled-side down. Grill until golden brown, 2 to 3 minutes. Carefully flip the quesadillas over and grill until the cheese has melted, 2 to 3 minutes longer.
Remove from the grill and cut each quesadilla into sixths. Top each wedge with a spoonful of Yellow Tomato-Tarragon Relish, a drizzle of Avocado Crema and serve.
Yellow Tomato-Tarragon Relish:
Combine the tomatoes, chile, onions, oil, vinegar and tarragon in a medium bowl. Season with salt and pepper.
Halve and pit the avocados. Scoop the flesh into the bowl of a food processor. Add the crema and puree until smooth. Season with salt and pepper.