Preheat the oven to 325°F. Place a muffin tin on a baking sheet and set aside.
Heat a medium skillet over medium heat with ½ tablespoon oil.
Add the ham and cook for 3 to 5 minutes until lightly browned. Remove from heat and let cool slightly.
Generously grease the muffin tin with the remaining oil.
In a large bowl, beat the eggs until fully combined and slightly frothy using a hand mixer or stand mixer.
With the mixer on low, beat in the Greek yogurt until smooth and fully combined.
Add the salt, pepper, onion powder, and paprika, and mix until the seasonings are evenly distributed.
Stir in ¾ cup of the shredded cheese.
Pour the egg mixture into the muffin tin, filling each cup about halfway.
Evenly distribute the cooked ham on top, then sprinkle with the remaining cheese. Gently tap the pan to settle the mixture.
Carefully pour hot water into the baking sheet (creating a water bath) until it reaches about halfway up the sides of the muffin tin.
Bake for 18 to 20 minutes, or until the egg bites are set in the center.
Remove from the oven and let cool for 5 minutes before removing from the tin.