Heat a sauté pan over medium-high heat.
Halve each hot cross bun and butter each side. Add to pan and let toast.
Flip each hot cross bun and add cheese on the bottom half. Let melt and add smoked ham. Then, remove buns from heat.
Wipe pan clean and add 1 tablespoon of oil and 1 tablespoon of butter. Crack the egg and gently cook sunny side up for about two minutes or until egg has set. Add extra butter and baste egg in order to help the egg set.
Add egg on top of ham and serve sandwich immediately.