Preheat oven to 350°F / 177°C.
Spray a 2 quart baking dish (or similar-sized dish, like a 9x9 pan) with nonstick cooking spray or brush with some oil.
In a small saucepan over medium heat, melt butter. Add flour and whisk well until no dry spots remain. Slowly pour in milk while whisking. Bring to a simmer and simmer until slightly thick, 2 to 3 minutes. Whisk in salt and black pepper.
Pour thickened sauce into prepared pan.
Scatter about half of the ham over the sauce.
Separate refrigerated biscuits and slice each biscuit into about 6 even pieces.
Spread half the biscuit pieces out in prepared pan.
Top with remaining ham and half the cheese.
Spread all remaining biscuit pieces on top.
Transfer baking dish to the oven and bake for 20 minutes. (The dough will not be fully cooked.)
Remove pan from oven. With the back of a spoon, make 6 deep wells in the biscuit mixture. Scatter remaining cheese into wells and over the rest of the dish. Crack an egg into each well.
Return pan to oven and continue baking until biscuits are golden brown and eggs are cooked to your liking, 10 to 15 minutes more. (The timing varies a bit depending on how deep you made the wells for the eggs and whether you’d like the yolks fully cooked or still slightly soft.)
Season the top of the eggs with some salt and pepper before serving.