Start by placing the pieces of day-old French bread in a bowl with the whole milk. Press on the bread so it absorbs the milk and softens it for about 10 minutes.
Squeeze out most of the milk from the bread, discard the excess liquid, then add the softened bread into a large bowl. Mix in the ground beef, eggs, onion, parsley, salt, pepper, and thyme until everything is well blended. Then shape into approximately 1¾-inch meatballs.
Preheat your oven to 350 °F (175 °C). Place the formed meatballs in a baking dish and bake for about 30 minutes. Let them cool slightly when done.
Dust the baked meatballs lightly with flour, shaking off any excess. In your skillet, melt the butter and olive oil over medium-high heat. Sauté the meatballs in two batches, until browned on all sides — this extra browning boosts flavor considerably.
Return all the browned meatballs into the skillet. Whisk the tomato paste into the red wine, add it, and bring to a boil for about 5 minutes until the liquid reduces slightly. Then pour in the beef broth, reduce the heat to medium, and simmer until the sauce thickens, about 15 minutes. Season with salt and pepper to taste.