Caramelize the onions. Melt 2 tablespoons of the butter along with the olive oil in a large dutch oven over medium heat. Once the butter is hot and foamy, add the onions and toss well to coat. Cover the pot and let the onions cook for 10 minutes, until they’re starting to wilt. Remove the lid, lower the heat, and add the thyme and some salt and pepper. Continue to cook the onions, stirring every so often, until they’re deeply caramelized. Stir in the dijon and balsamic and set aside.
Mix the burger meat. Transfer half of the cooled caramelized onions to a cutting board and finely chop them. Transfer the chopped caramelized onions to a large mixing bowl with the ground beef, Worcestershire, dijon, dehydrated onion, garlic powder, salt and pepper. Use your hands to gently mix the meat so that everything is combined, then divide into 4 - 6oz balls. Form the meat into patties at least 1 inch thick, and a bit bigger than your burger bun (they’ll shrink on the grill). Use your thumb to press an indention into the center of each patty.
Make the burger sauce. In a small bowl, mix together the mayo, dijon, horseradish, and honey and season with salt and lots of freshly cracked black pepper.
Prep your grill. Preheat an outdoor gas or charcoal grill until it reaches 450 F. Make sure the grates are brushed clean of debris. Lightly oil the grates to keep the burgers from sticking.
Grill the burgers. Grill the burgers over direct heat for 3-4 minutes per side. Remove the burgers from the grill and let them rest. While they’re still warm, top the patties with the slices of brie.
Assemble the burgers. Spread some of the burger sauce on the toasted buns and top with the burger patties. Divide the rest of the caramelized onion mixture between the burgers and top with arugula. Serve right away.