For the caramelized onions: Place a large, heavy bottomed pot over medium heat. Add the olive oil and butter and heat until the butter melts. Add the onions, thyme sprigs, bay leaf and garlic. Sprinkle the onion mixture with 1 teaspoon salt and 1/2 teaspoon pepper and cook over medium heat, stirring frequently, until the onions are soft and golden brown, 45 minutes.
For the arugula aioli: Meanwhile, add the mayonnaise, arugula, lemon juice and 1/4 teaspoon kosher salt to the carafe of a high-speed blender. Blend on high speed until it becomes a smooth sauce that is vibrant green in color, about 30 seconds. Set aside for sandwich assembly or store, refrigerated, in an airtight container for up to 2 weeks.
For the panini: Lay 2 pieces of bread out on a work surface. Spread each piece of bread with 1 tablespoon of arugula aioli. Place 3 slices of Brie on 1 piece of bread. Layer the roast beef over the Brie. Spread 3 tablespoons of caramelized onions over the beef. Close the sandwich with the other piece of bread, aioli facing in. Repeat this process to assemble 3 more sandwiches.
Place a large saute pan over medium heat. Gather a piece of parchment paper and a cast-iron pan to use as a weight on top of the sandwiches.
Pour 1 tablespoon olive oil into the pan, followed by 2 sandwiches. Lay the parchment over the sandwiches and cover with the cast-iron pan. Gently press the cast iron into the sandwiches and cook until the bread is golden brown and crispy, about 2 minutes. Remove the cast-iron pan and parchment and flip the sandwiches over. Replace the parchment and cast-iron pan and cook the sandwiches until the bread is toasted and the cheese begins to melt, about 2 minutes more. Remove the sandwiches from the pan and repeat this process with the remaining tablespoon of olive oil and 2 sandwiches.