Make the Panade: In a large mixing bowl, combine the milk (1/2 cup) and Panko (3/4 cup). Let it sit for 10 minutes, then mash with a fork into a paste.
Mix the Meatballs: Add ground beef (2 pounds), onion (1/2 cup), garlic (4 cloves), egg (1), Dijon (1 tablespoon), Worcestershire (1 tablespoon), Italian seasoning (2 teaspoons), thyme (1/2 teaspoon), salt (1 teaspoon), and pepper (1/2 teaspoon). Mix gently with your hands until just combined. (Don’t overmix or they’ll get tough.)
Shape: Roll into 1 1/2–2-inch meatballs, compressing enough that they hold together, and set on a tray. Refrigerate for 15 minutes to firm up.
Brown the Meatballs: Heat 2 tablespoons of oil in a large skillet. Working in batches, add the meatballs, leaving space between each. Cook without moving for 3–5 minutes, until browned on the bottom, then carefully flip and brown the other side. Transfer to a plate and tent loosely with foil. Add the remaining tablespoon of oil as needed between batches and continue until all the meatballs are browned.
Start the Gravy: In the same skillet, melt the butter (1/4 cup). Add the mushrooms (8 ounces) and onions (1) and cook over medium-low heat for 10–12 minutes, stirring occasionally, until the onions are softened and lightly golden and the mushrooms are browned. Adjust the heat to low as needed so the vegetables soften slowly without over-browning. Stir in the garlic (1 tablespoon) and cook for 1 minute more, just until fragrant.
Build the Roux: Sprinkle flour (1/4 cup) over the onion mixture, stirring until it forms a paste. Cook for 1–2 minutes to remove the raw flour taste.
Add Liquid: Gradually whisk in the beef broth (3 1/2 cups), Worcestershire sauce (2 teaspoons), Better Than Bouillon (2 teaspoons), and black pepper (1/4 teaspoon). Bring to a gentle boil, stirring until the gravy thickens. Taste and add salt only if needed.
Finish Cooking: Nestle the browned meatballs back into the gravy with any accumulated juices. Reduce heat to low and simmer for 10–15 minutes, turning once halfway through, until the meatballs are fully cooked (160°F on an instant-read thermometer).