Mash garlic and ginger in a mini food processor or mortar and pestle until smooth. Add remaining dressing ingredients and mix until smooth, tasting for seasoning and heat. Set aside.
Cut green beans in half and place in a large skillet with water to cover. Bring water to a boil and cook until beans are just tender and bright green. Drain and shock with cold water to retain color. Place in a large serving bowl.
Meanwhile, cut peppers into thin strands. Add to sserving bowl. Remove ends from cucumbers and cut in half, then slice into thin pieces to match shape of peppers. Add to salad bowl. Cut carrots into matchsticks and slice scallions and add both to salad bowl.
Rinse mushrooms and remove dirt, then slice in narrow slices. Heat vegetable oil over medium heat and add mushrooms, sesame oil, and soy sauce. Cook until mushrooms are cooked through and add hoisin sauce to coat. Add mushrooms to serving bowl and toss, then add dressing and toss to coat evenly. Serve and garnish each plate with sesame seeds.