Preheat the oven to 375°F (190°C). Lightly grease a 12-cup muffin tin with nonstick spray or a thin swipe of oil. Even if your pan is nonstick, give it a quick coat. It’s worth it.
If you’re adding veggies, warm a skillet over medium heat with the olive oil or butter. Sauté the chopped veggies for 3 to 5 minutes until softened and most of the moisture cooks off. Set aside to cool slightly. This step keeps the egg cups from turning watery.
Make the ham “cups.” Press a slice of deli ham into each muffin well so it forms a little bowl. The edges can fold and ruffle a bit, that’s fine. If your ham is very thin, overlap two half slices to cover the bottom. If you’re using diced ham instead, divide it evenly among the cups and skip the lining step.
Whisk the eggs in a medium bowl with the milk, salt, pepper, and paprika or garlic powder if using. Whisk just until the yolks and whites combine. No need to go wild here.
Divide the shredded cheese among the cups, keeping a little back for the top if you like a cheesy cap. Add a spoonful of the sautéed veggies to each cup (totally optional, but good).
Pour the egg mixture into each ham cup, filling about three-quarters of the way. The eggs will puff a bit in the oven. Sprinkle on the last bit of cheese if you saved some.
Bake for 14 to 18 minutes, until the centers are just set and no longer jiggly. If you like precision, an instant-read thermometer should read around 165°F (74°C) in the center. They’ll continue to firm up slightly as they rest.
Let the cups cool in the pan for 5 minutes, then run a small knife around the edges and lift them out. Garnish with chopped chives or parsley if you’re feeling fancy. Serve warm with a dab of hot sauce or a spoonful of salsa.