Combine peaches, mayonnaise, tarragon, and 1 teaspoon chile-garlic sauce in the bowl of a food processor; process until smooth, about 1 minute. Place aioli in the refrigerator until ready to use.

Preheat an outdoor grill for medium-high heat and lightly oil the grate.

Mix ground pork, applesauce, bread crumbs, 1 teaspoon chile-garlic sauce, salt, garlic powder, onion powder, and pepper in a bowl. Divide pork mixture into 4 equal portions and form into burgers.

Place burgers on the preheated grill and cook until an instant-read thermometer inserted into the centers reads at least 160 degrees F (70 degrees C), about 3 minutes per side.

Spread aioli onto buns and top with burgers.