Light the barbecue and set for direct/indirect cooking.
On the stovetop, soft-boil the eggs and then immediately transfer into ice-cold water to arrest the cooking process.
Place a small saucepan over medium heat on the stovetop and add the butter, olive oil and shallots. When the butter starts to bubble, turn the heat down to low and cook the shallots very gently for about an hour or until very tender and lightly caramelized. Season to taste, then add the vinegar and keep warm.
Carefully peel the eggs and place one in each hole of the muffin tin. Place the muffin tin in the indirect heat zone, then sprinkle a good handful of wood chips directly onto the coals. Close the lid and vent of the barbecue and leave the eggs to smoke to 3 minutes. Open the lid, carefully turn the eggs and add another handful of wood chips, then close the lid again and cook for another 3 minutes. Lift the lid to check the eggs – they should have smoked to a brown-orange hue. Move them to the indirect heat zone to keep warm.
Toss the asparagus in olive oil and season well. Place on the grill in the direct heat zone and cook for 2–3 minutes on each side until just tender and lightly charred. Transfer to the indirect heat zone, then grill the bread slices in the direct heat zone until lightly browned.
Spoon the caramelized shallots onto the grilled bread, then top with the asparagus and a smoked egg. Just before serving, break each egg gently with a fork and season with a little salt and pepper.