In a small saucepan, bring ginger, chile, sugar, and 1/3 cup water to a boil. Reduce heat to low; cook until ginger and chile are soft, about 5 minutes. Use a slotted spoon to transfer ginger and chile to a bowl; set aside. Reserve syrup.
Make dressing: In a shallow bowl, whisk together lime juice, soy sauce, and 2 teaspoons reserved syrup. Add tofu, and toss to coat. Set aside.
In a pot of boiling water, cook the noodles according to package instructions. Drain; transfer to a large bowl. Add peas; drizzle with oil and 1 tablespoon dressing. Toss to coat; let cool.
To serve, add cucumber and chives to bowl along with the tofu and dressing; toss to combine. Divide among plates; top with avocado and reserved ginger and chile. Sprinkle with sesame seeds.