Roast the garlic. Preheat the oven to 375°F. Slice the tops off both garlic heads to expose the cloves. Place each on a piece of foil, drizzle each with 1 teaspoon olive oil, sprinkle with salt and wrap tightly. Roast for 40–50 minutes, or until the cloves are very soft and golden. Let cool enough to handle, then squeeze out the cloves. Increase the oven temperature to 425°F for the meat.
Caramelize the onions. While the garlic roasts, heat the butter and olive oil in a very large, wide skillet or Dutch oven over medium heat. Add the onions and salt, then reduce the heat to medium-low. Once the onions have released their liquid add in 1/4 cup of broth or water and the balsamic vinegar and continue to cook, stirring occasionally, for about 45–75 minutes, or longer as needed, until deeply golden and soft. Let cool, then finely chop ½ cup for the meat mixture. Reserve the rest for topping.
Mix the filling. Line a rimmed baking sheet with foil and lightly grease it. In a large bowl, combine the panko and milk; let stand for 2–3 minutes. Stir in the egg, soup mix, ½ cup finely chopped caramelized onions, Worcestershire, Dijon, and thyme. Add the ground beef and mix gently just until combined.
Shape and bake. Shape the mixture on the prepared baking sheet into an even rectangle or oval slightly larger than your ciabatta, about 1 inch thick. Bake at 425°F for about 20–25 minutes, or until a thermometer inserted into the thickest part reads 160°F. Continue baking as needed to reach that temperature. Rest for 5–10 minutes.
Make the Dijonnaise. While the meat bakes, mash the roasted cloves from both heads into a paste. Stir together with the mayonnaise, Dijon, and Worcestershire. Season with salt and pepper to taste. Add more Dijon if desired.
Assemble. Set the oven to broil. Place both ciabatta halves cut side up on a clean baking sheet. Spread the bottom half generously with Dijonnaise. Using two wide spatulas, lift the meat onto the bottom half, leaving the rendered fat behind. Top with the remaining caramelized onions and Gruyère. Drizzle the cut side of the top half of the bread with olive oil.
Broil. Broil for 1–2 minutes, watching closely, until the cheese melts and the top half of the bread is lightly toasted. Remove the top half early if needed.
Finish and serve. Spread more Dijonnaise over the toasted top half. Top the melted cheese with arugula, close the sandwich, and slice into portions. Serve warm.