In a small bowl, add butter and 3 Tbsp of blue cheese crumbles. Mash with a fork to blend. Set aside.
In a mixing bowl, combine ground beef, seasoning and remaining blue cheese. Form into four patties, cover with plastic wrap and chill in the refrigerator for an hour to set.
Preheat BBQ grill, then add patties, sprinkling salt and pepper on top. Grill for 4-5 minutes then flip over, cooking another 4-5 minutes to reach a medium doneness. Use a BBQ thermometer to check temperature for accuracy.
Remove burgers and let them rest for a few minutes.
Place arugula on bottom buns, add burger then spread on blue cheese topping. Serve immediately.
Optional: Substitute Gorgonzola if desired, or add crumbled bacon to burger mix before grilling.