Prepare the hamburger mixture: In a large bowl, grate the onion. Add ground beef, breadcrumbs, salt, black pepper, heavy cream, and egg. Mix gently by hand until just combined. Avoid overmixing to keep the patties tender.
Form the patties: Divide the beef mixture into four equal portions. Shape each into a ball, then press each ball to about 3/4 inch (2 cm) thick. Create a slight indentation in the center of each patty to prevent it from puffing up during cooking.
Prep the sauce ingredients: Finely chop the remaining onion, mince the garlic, and chop the dill for the sauce. Having these ingredients ready will make the sauce preparation smoother once the patties are cooked.
Cook the hamburger steaks: Heat olive oil and butter in a large nonstick skillet over medium heat. Add the formed patties and cook for 4–5 minutes per side until golden brown and caramelized. Transfer cooked patties to a clean plate and set aside.
Start the sauce base: Reduce heat to low. In the same skillet, add diced onion and minced garlic. Cook for 2 minutes until fragrant and translucent, being careful not to burn the garlic.
Create the roux: Add salt, black pepper, and flour to the skillet. Stir continuously until the flour is incorporated and cooked, about 1 minute. This will prevent any raw flour taste in the final sauce.
Build the sauce: Pour in the chicken broth, scraping up any browned bits from the bottom of the pan using a wooden spoon. These fond bits contain concentrated flavor. Simmer for 2 minutes to begin thickening the sauce.
Finish the sauce: Add heavy cream and chopped dill to the skillet. Bring to a boil, then reduce heat and simmer 1–2 minutes until the sauce thickens to coat the back of a spoon. The dill will infuse the sauce with its aromatic flavor.
Combine and simmer: Return hamburger steaks to the skillet, nestling them into the sauce. Simmer in the sauce for 5–6 minutes, turning occasionally, until heated through and cooked to desired doneness in the center.
Garnish and serve: Finish with extra chopped fresh dill and freshly ground black pepper. Serve hot, ideally with mashed potatoes to soak up the delicious sauce.