In a small saucepan, heat the water, sugar, vinegar, and salt. Place the carrots and cucumbers in a heatproof bowl, cover with the vinegar mixture and cool to room temperature.
Preheat the oven to 400°F.
On a baking sheet, toss the yams with the oil and a big pinch of salt and pepper. Roast for 15-20 minutes, until tender.
In a medium saucepan, heat the soy sauce, water, sugar, scallions, garlic, sake, dried shiitakes, and ginger. Simmer for 20 minutes, then use a strainer to remove the vegetables.
In a small bowl, combine the cornstarch with 1 tablespoon water to form a paste. Whisk the cornstarch slurry into the teriyaki sauce until thickened. Remove from the heat.
Meanwhile, in a small saucepan, bring the rice, 2 cups of water and a pinch of salt to a simmer. Reduce heat to low, cover, and cook until the grains are tender, about 20 minutes.
On a baking sheet, toss the corn with the olive oil and a pinch of salt and pepper and roast for 10 minutes. Remove from the oven and toss with the edamame.
Toss the roasted yams with the teriyaki sauce and scallions to coat.
Divide the rice between bowls. Top with teriyaki yams, corn and edamame, radishes, sesame seeds, garlic chips and cilantro.