For the dressing, in a screw-top jar combine olive oil, vinegar, thyme, and pepper. Cover and shake well. Set aside.
In a large bowl combine ground beef, 1/4 teaspoon salt and 1/4 teaspoon ground black pepper. Shape ground beef into four 3/4-inch-thick patties. Brush squash slices with some of the dressing mixture. Set remaining dressing aside.
For a charcoal grill, grill patties on the rack of an uncovered grill directly over medium coals for 14 to 18 minutes or until meat is done (160 degrees F), turning once halfway through grilling. Place squash slices along side beef patties. Cook over medium coals for 7 to 10 minutes or until tender, turning once. (For a gas grill, preheat grill. Reduce heat to medium. Place patties on grill rack over heat. Cover and grill as above.)
To serve transfer cooked beef patties to buns. Drizzle cut side of bun tops with remaining dressing. Top burgers with arugula, squash, blue cheese, tomato, red onion, and pickle slices. Makes 4 servings.