In a large bowl gently combine the ground beef, salt, pepper, Worcestershire sauce and liquid smoke. Divide ground burger mixture into 4 equal segments and form them into thick patties. Refrigerate on a plate for 1 hour, then remove to and allow to come to room temperature, about 30 minutes before cooking.
In a small bowl, mix together the garlic, mayonnaise and lemon juice and salt, to taste. Refrigerate until ready to serve.
Warm a grill pan over high heat. Lightly brush the top of each burger with vegetable oil, then arrange on the grill, oiled side down. Sear until the burgers release easily from the grill and are brown, about 5 minutes. Brush with a bit more vegetable oil, flip and cook until the center slightly springs back, indicating a medium-rare center, another 5 minutes. While the burgers are finishing, brush both sides of the bread lightly with olive oil then put on the grill and cook until toasted, about 1 minute, then flip and toast the other side.
Transfer 1 slice to each serving plate and spread with the garlic spread. Top each slice with a burger and while they are resting make the eggs to top each burger. In a small pan, melt 1 tablespoon of butter over low heat, crack an egg in the center, cover and cook until the white is set and the yolk is still runny, about 5 minutes. Slide the egg on top of the of the burger, season with a bit of salt and pepper, then finish with a few shakes of hot sauce. Repeat with the remaining butter and eggs and serve.