Preheat the grill to medium low heat. Brush the mushroom cap, endive pieces, and radicchio pieces with the avocado oil using a basting brush. Please them on the grill and cook for approximately 5 minutes, turning half way through cooking until all vegetables become tender.
While the vegetables are cooking, place the arugula and cucumber in a large bowl.
In a separate small bowl, combine the balsamic, olive oil, and mustard, and mix well.
Place the grilled vegetables on top of the arugula and cucumber mix, dress, mix, and serve.