In a small bowl stir together ground coffee, brown sugar, paprika, 1 1/2 teaspoons black pepper, kosher salt, and thyme; set aside.
In another small bowl combine ketchup, horseradish, and Worcestershire sauce. Season to taste with hot pepper sauce; set aside.
In a small skillet heat oil over medium heat. Add shallots; cook for 3 to 4 minutes or until lightly browned and crisp, stirring frequently. Using a slotted spoon, remove shallots and drain on paper towels. Sprinkle with salt and additional black pepper.
Shape ground beef into four 1/2-inch-thick patties. Generously coat both sides of patties with spice mixture, pressing into meat to adhere.
For a charcoal or gas grill, place patties on the rack of a covered grill directly over medium heat. Grill for 10 to 13 minutes or until done (160 degrees F), turning once halfway through grilling and adding cheese slices to patties for the last 1 minute of grilling.
To serve, arrange lettuce and pickle slices on bottoms of rolls; add burgers. Top with fried shallots and ketchup mixture. Replace tops of rolls.