Rinse the beef liver slices under cold water and pat dry with paper towels.
In a shallow dish, combine the flour, salt, black pepper, and paprika.
Dredge each slice of liver in the flour mixture, coating both sides evenly. Shake off any excess flour.
Heat 2 tablespoons of olive oil and 2 tablespoons of butter in a large skillet over medium-high heat until the butter is melted and foaming.
Add the liver slices to the skillet in a single layer. Cook for 2 to 3 minutes on each side until browned but still slightly pink inside. Avoid overcooking to keep the liver tender.
Remove the liver from the skillet and set aside on a plate, covering loosely with foil to keep warm.
In the same skillet, add the remaining 1 tablespoon of butter. Add the sliced onions and cook over medium heat, stirring occasionally, until they are soft and caramelized, about 7 minutes.
Add the minced garlic to the onions and cook for an additional 1 minute until fragrant.
Pour in the beef broth to deglaze the skillet, scraping up any browned bits from the bottom. Let the broth simmer for 2 minutes until slightly reduced.
Return the liver to the skillet, spoon some of the onion and broth mixture over the slices, and heat through for 1 minute.
Sprinkle the chopped fresh parsley over the liver and onions before serving.