Clean the chanterelles and quickly rinse under running water. Dry thoroughly with paper towels. Cut larger mushrooms, leave smaller ones whole. Prepare a double portion of balsamic sauce and mix it with honey and mustard.
Preheat a grill pan, cut the bell pepper into pieces and remove the seed pods. Sprinkle with salt and coat with olive oil. Arrange on the pan, cover with aluminum foil and grill 2-3 minutes on each side, until the skin is lightly brown, season with freshly ground black pepper. The pepper can also be fried in a regular pan or placed under a preheated grill in the oven.
Meanwhile, heat a non-stick pan well, pour in olive oil, add chanterelles, salt and fry for a few minutes, stirring occasionally. Halfway through frying, add butter. Remove from heat, sprinkle with dill and season with pepper.
Arrange arugula on plates, sprinkle with salt and season with pepper. Cut the avocado in half, remove the pulp with a spoon and cut into pieces. Arrange chanterelles, pepper and avocado on the arugula. Sprinkle with Parmesan shavings and drizzle with balsamic sauce. Serve with bread.