Mince the onion and microwave it in a heatproof container covered with plastic wrap for 30 seconds at 600W. Drain and let it cool.

In a bowl, soak breadcrumbs in milk. Add all other meat mixture ingredients except vegetable oil and mix until sticky.

Shape the meat mixture into oval patties, pressing to remove air and creating a slight indentation in the center.

Prepare the egg mixture by whisking eggs in a bowl. Add milk and salt, and whisk again.

Heat vegetable oil in a pan over medium heat. Cook the patties until browned on one side, flip, cover, and cook on low heat for 7–8 minutes. Remove and clean the pan.

In the cleaned pan, melt butter over medium heat. Pour in the egg mixture and stir gently while cooking.

Place the cooked patties in the center of the eggs, fold the eggs over, and transfer to a plate.

In the same pan, prepare the demi-glace sauce by combining all sauce ingredients and heating over medium heat until warm.