Mix ground turkey, minced red onion, Dijon mustard, and coarse salt in a bowl; form into 4 patties and refrigerate 30 minutes.
Preheat an outdoor grill for medium-high heat and lightly oil the grate.
Drizzle olive oil over the burgers.
Cook burgers on the preheated grill until no longer pink in the center and the juices run clear, 4 to 5 minutes per side. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).
Serve on whole wheat hamburger buns topped with baby arugula, avocado, green bell pepper, and Parmesan cheese.