Add Summer Ale, garlic, paprika, crushed red pepper, thyme, bay leaf, salt, and oil to a shallow baking dish. Mix until salt has dissolved. Add chicken and lemons, cover, and refrigerate for at least 4 hours, up to overnight.
Heat a grill for direct heat. Add chicken, discarding marinade, and grill for 6-8 minutes per side. Add cheese during last minute of cooking. Internal temperature of chicken should read 165F. Let rest for 10 minutes.
Meanwhile, lightly oil and season red onion slices. Grill over direct heat, turning, until nicely browned, about 10 minutes.
To make dressing, mix mayo, honey, lemon juice, and hot sauce in a small bowl. Season to taste.
Assemble chicken sandwiches by layering tomatoes, lettuce, dressing, grilled chicken, grilled onions, and a little more dressing on buns.