Preheat your oven to 350°F and generously grease a standard 12-cup muffin tin with butter or cooking spray, making sure to coat the bottom and sides so the muffins release easily. While the oven heats, dice the ham into small 1/4-inch cubes and thinly slice the green onions using only the white and light green parts, discarding the dark green tops which can become bitter when baked.
In a large bowl, whisk together the 13 eggs and milk until well combined and slightly frothy. Add the salt, black pepper, garlic powder, and smoked paprika, whisking to distribute the spices evenly throughout. Fold in the diced ham, sliced green onions, and 1/2 cup of the cheddar cheese until everything is evenly distributed. I like to reserve the best cheese for topping because it gets golden and crispy in the oven, which adds great texture to the finished muffins.
Divide the egg mixture from Step 2 evenly among the 12 prepared muffin cups, filling each about three-quarters full. Top each muffin with a generous pinch of the remaining 1/4 cup cheddar cheese, distributing it evenly so each muffin gets a nice cheesy crust. I find that freshly shredded sharp cheddar melts more evenly than pre-shredded varieties, creating a better texture on top.
Place the muffin tin in the preheated 350°F oven and bake for 25 to 30 minutes, until the egg muffins are set and no longer jiggle in the center when gently shaken. The tops should be lightly golden and the cheese should be melted and starting to brown at the edges. Start checking around the 25-minute mark to prevent overbaking, which can make them rubbery.