Combine 1/2 cup blue cheese, mayonnaise, 1 teaspoon thyme, and hot pepper sauce in a small bowl; stir well.
Divide beef into 4 equal portions, shaping each portion into a 1/2-inch-thick patty. Sprinkle beef evenly with 1/2 teaspoon black pepper and salt.
Brush oil evenly over both sides of onion slices; sprinkle with remaining 1/2 teaspoon pepper. Place the patties and onions on grill rack coated with cooking spray; cover and grill for 3 minutes on each side. Set patties aside; keep warm. Place onion slices in a zip-top plastic bag; seal. Let stand 5 minutes; toss with remaining 1 teaspoon thyme and vinegar.
Spread cut sides of buns evenly with mayonnaise mixture. Arrange 1/2 cup arugula on bottom half of each bun; top with 1 patty, 1 onion slice, and bun top.