Preheat grill to 350˚F (180˚C).
Make foil ‘boats’ for avocados to sit in by folding up sides of four 8 x 6” (20 x 15 cm) pieces of aluminum foil while leaving the base flat.
Cut avocados in half lengthwise and remove pits.
Scoop out avocado ‘meat’ with a spoon, leaving 1/4” (1 cm) around the sides.
Roughly mash avocado ‘meat’ with a spoon. Fold in smoked ham. Season lightly with lime juice, salt and pepper.
Spoon mixture back into avocado shells. Smooth it out so filling is flush with the top of each shell.
With the back of the spoon, create a well in each avocado half large enough for an egg to fit into. You may need to remove some of the filling (reserve as garnish).
Place avocado halves onto foil boats. Crack an egg into a small bowl, then gently transfer into the avocado. Be sure not to break the yolk. Repeat with remaining eggs and avocado halves.
Carefully place avocados and eggs onto the grill. Cook with lid down until whites are set, approximately 15 to 20 minutes.
Finish with a pinch of salt and pepper and any reserved avocado and ham mixture.