If you love cheeseburgers so much, these gruyere and egg burgers are a must-try. It's loaded with different ingredients that complement one another so so much.
Whisk lemon zest, lemon juice, garlic, and mayonnaise together until combined. Let sit in the refrigerator.
For making burgers, mix the next 6 ingredients together, tossing lightly until thoroughly mixed (make sure not to overmix).
Form into 8 patties. Cover and grill over medium direct heat for 5 to 7 minutes per side until a thermometer reads 160 degrees. Use cheese to dredge on top; cover and keep grilling for 1 to 2 more minutes until cheese melts. On bun bottoms, place burgers. Keep warm.
In each of 2 large skillets, melt a tablespoon of butter over medium heat (on stovetop or grill). Break eggs into a saucer or a custard cup, one egg at a time, then slide into pans gently. Lower heat to low right away. To make eggs sunny-side up, cook, cover, till yolks get thickened yet are not hard.
For making basted eggs, during the cooking process, in the pan, spoon butter over the eggs. For making over-easy, flip eggs carefully for both sides to cook, yet do not cover the pan.
For serving, spread over bun tops with mayonnaise mixture.
Place tomatoes, arugula, and extra mustard (optional) onto burgers. Top burgers with fried eggs. Replace bun tops.