How to Make It
In a large skillet, heat 1 tablespoon of olive oil until shimmering. Add the onion rings, season with salt and pepper and cook over high heat until richly browned, about 2 minutes per side. Transfer the onion rings to a medium bowl.
Add 3 tablespoons of the olive oil to the skillet and when it is hot, add the chanterelles and season with salt and pepper. Cook over moderate heat until the liquid in the skillet has evaporated, about 7 minutes. Continue to cook over moderate heat, stirring occasionally, until the chanterelles are browned, about 5 minutes longer. Season with salt and pepper.
In a small bowl, combine the vinegar with the mustard and the remaining 2 tablespoons plus 1 teaspoon of olive oil; season the vinaigrette with salt and pepper. Add the arugula, blackberries and raspberries to the bowl with the onions. Add the chanterelles and the vinaigrette and toss well. Put a piece of cheddar on each plate. Divide the salad among the plates and serve.