For the tarragon aioli: In a bowl, whisk together the mayo, garlic puree and tarragon. Taste and season with salt and white pepper as desired.
For the chicken burgers: Preheat the oven to 350 degrees F. Over medium-high heat, warm a large ovenproof saute pan with the oil. Form the chicken into 4 equal patties, and then form a pocket in the center of each burger. Fill the center with a quarter of the blue cheese, and then fold the edges to seal the cheese in the burgers. Sprinkle the stuffed burgers with the salt and pepper and sear in the pan for 2 to 3 minutes per side. Transfer the pan to the oven and finish cooking until the internal temperature reaches 165 degrees F.
Toast the buns with the butter on a grill, and then layer the toasted bun bottom with some tarragon aioli, baby spinach leaves, a tomato slice, burger and top of bun. Repeat with the remaining ingredients.