Heat a large non-stick frying pan over medium-high. Cook bacon until crispy, 2 to 3 min. Transfer to a paper-towellined plate. Set aside.
Preheat barbecue to medium. Whisk egg with Dijon, Worcestershire and salt in a large bowl. Season with fresh pepper. Add beef. Gently mix together. Shape into 4 burgers, each about 1/2 in. thick. Press a shallow depression in centre of each to help keep their shape while cooking.
Oil grill and barbecue burgers, lid closed, 5 to 7 min. Flip burgers, then lay smoked-cheddar slices on burgers. Continue grilling, lid closed, until burgers are cooked and cheese is melted, 6 more min.
Place bottom buns, cut-side up, on grill. Lay aged-cheddar slices on buns. Grill, lid closed, until cheese is melted, 2 to 3 min. Stack beef patty, tomato slice, bacon, arugula and Lemon-Pepper Aioli between buns.