Put the chicken breasts in a food processor or blender and process until smooth. Add the breadcrumbs, tarragon and mustard and process again. Shape the mixture into 4 balls and flatten them into 4 patties with lightly oiled hands.
Heat the oil in large, heavy-based griddle pan or frying pan and cook the burgers, turning once, over a moderately high heat for 8–10 minutes or until golden and cooked through.
Split the buns and serve the burgers in the buns with slices of Brie and a spoonful of spicy tomato salsa.