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Preheat grill to 350˚F (180˚C).
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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.
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Cut avocados in half lengthwise and remove pits.
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Scoop out avocado ‘meat’ with a spoon, leaving 1/4” (1 cm) around the sides.
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Roughly mash avocado ‘meat’ with a spoon. Fold in smoked ham. Season lightly with lime juice, salt and pepper.
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Spoon mixture back into avocado shells. Smooth it out so filling is flush with the top of each shell.
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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).
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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.
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Carefully place avocados and eggs onto the grill. Cook with lid down until whites are set, approximately 15 to 20 minutes.
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Finish with a pinch of salt and pepper and any reserved avocado and ham mixture.