Whisk mayonnaise, garlic, lemon juice and lemon zest until blended. Refrigerate.
For burgers, combine the next six ingredients, mixing lightly but thoroughly (do not overmix). Shape into eight patties. Grill, covered, over medium direct heat until a thermometer reads 160°, 5-7 minutes on each side. Top with cheese; grill, covered, until cheese is melted, 1-2 minutes longer. Place burgers on bun bottoms. Keep warm.
Melt 1 tablespoon butter over medium heat in each of two large skillets (on grill or stovetop). Break eggs, one at a time, into a custard cup or saucer, then gently slide into pans. Immediately reduce heat to low. To prepare eggs sunny-side up, cover pan and cook until yolks thicken but are not hard. To make basted eggs, spoon butter in pan over eggs while cooking. For over-easy, carefully turn eggs to cook both sides but do not cover pan.
To serve, spread mayonnaise mixture over bun tops. Add arugula, tomatoes and, if desired, additional mustard to burgers. Top with fried eggs. Replace bun tops.
Test Kitchen tips
Want big burgers? Just make 4 patties instead of 8; you can cut the eggs down to 4 as well.
For extra springy flavor, mix fresh dill fronds into the burger patties instead of thyme.
For a healthier alternative, use ground turkey in place of ground beef.