Preheat oven to 450 degrees F (230 degrees C). Line a baking sheet with foil.

Melt butter in a small saucepan over medium heat. Stir in brown sugar, and cook until the sugar has melted and turned deep brown - but not burnt, 5 to 8 minutes.

Once the sugar has browned, stir in walnut halves until coated with caramelized sugar. Spread out onto prepared baking sheet, and place into preheated oven. Cook until the center of the walnuts have just turned a light golden brown, about 5 minutes. Once done, remove from the oven and allow to cool until warm or room temperature.

Whisk the vinegar together with the chives, parsley, dill, and chervil. Whisk in walnut and peanut oils until well combined.

To serve, toss the arugula with enough dressing to moisten in a large bowl. Season to taste with salt and pepper. Place onto salad plates and garnish with candied walnuts.