Preheat the oven to 450°F. Line a sheet pan with foil. Wash and dry the fresh produce. Peel the carrots; halve lengthwise, then cut crosswise into 2-inch pieces. Transfer to the sheet pan. Drizzle with olive oil; season with salt and pepper. Toss to coat; arrange in an even layer. Roast 16 to 20 minutes, or until browned and tender when pierced with a fork. Remove from the oven.

Meanwhile, grate the gouda on the large side of a box grater. Halve, peel, and thinly slice the onion(s). In a bowl, combine the mayonnaise and half the worcestershire sauce. Taste, then season with salt and pepper if desired.

In a medium nonstick pan (or large for 4 servings), heat a drizzle of olive oil on medium-high until hot. Add the sliced onion(s). Cook, stirring occasionally, 6 to 8 minutes, or until lightly browned and softened. Carefully add the honey, hot sauce, and remaining worcestershire sauce. Cook, stirring frequently, 1 to 2 minutes, or until combined and the onion is coated. Transfer to a bowl. Taste, then season with salt and pepper if desired. Cover with foil to keep warm. Rinse and wipe out the pan.

In the same pan, heat a drizzle of olive oil on medium-high until hot. Add the patties. Cook 4 to 6 minutes, or until browned. Flip the patties and evenly top with the grated gouda. Loosely cover the pan with foil. Cook 4 to 6 minutes, or until the cheese is melted and the patties are heated through. Transfer to a work surface.

In the same pan, heat a drizzle of olive oil on medium-high until hot. Working in batches if necessary, add the buns, cut side down. Toast 30 seconds to 1 minute, or until lightly browned. Transfer to a work surface. Assemble the burgers using the toasted buns, cooked patties, and caramelized onion(s). Serve the burgers with the roasted carrots and sauce on the side. Enjoy!