Preheat grill to medium heat.
In a small bowl, combine soy sauce, mirin, and sugar. Brush eel fillets with mixture.
Grill eel fillets for 5-6 minutes on each side, or until cooked through.
Meanwhile, heat vegetable oil in a large skillet over medium heat. Add carrot, bell pepper, snow peas, and green onions. Cook for 5-6 minutes, or until vegetables are tender.
Cook soba noodles according to package instructions. Drain and rinse with cold water.
In a small bowl, whisk together sesame oil and rice vinegar. Toss soba noodles with dressing.
Divide soba noodles between two bowls. Top with grilled eel fillets and sautéed vegetables. Sprinkle with sesame seeds.
Serve and enjoy!