Melt olive oil and butter in a large nonstick pot over medium heat. Add onions and stir with a wooden spatula until translucent, about 3 minutes. Add flour and stir to create a thick paste, about 2 minutes. Pour in milk while stirring; it will seem lumpy at first, but it will smooth out to the consistency of mashed potatoes, about 5 minutes. Add nutmeg; keep stirring until a dough forms and starts to pull away from the sides of the pot, about 5 minutes more.

Stir ham into the pot and mix into the dough until well combined, about 2 minutes. Add parsley. Transfer dough into a shallow bowl and allow it to cool.

Place eggs in a bowl and beat. Place bread crumbs in a separate bowl.

Oil the palm of 1 hand; scoop 2 tablespoons of dough into the oiled hand using a silicone spatula. Shape the dough into an oblong capsule. Place the croquette into the bowl with the eggs. Use a fork to transfer it to the bowl with the bread crumbs; roll using fingers to coat well. Transfer to a plate; repeat with remaining dough, placing croquettes side-by-side.

Heat 1 1/2 inches oil in a large skillet over medium heat. Add croquettes to the skillet without crowding and brown on all sides, 6 to 8 minutes. Place on a paper towel to absorb excess oil.