Whisk the sauce ingredients together in a bowl until smooth. Set aside.
Prepare the vegetables
Toss together the eggplant, shiitake, bell pepper, tomatoes, onion, olive oil, chili pepper and salt on a large rimmed baking sheet. Roast 20-25 minutes, until the vegetables are tender and the eggplant is golden brown
Add the green onion and spinach to the pan and stir until the leaves wilt.
Prepare the udon noodles according to package directions. Pour half the sauce over the noodles and toss with tongs until they're coated.
Portion the noodles into bowls and top with the vegetables. Drizzle the remaining sauce over and serve.
The roasted vegetables can be made 5 days ahead. Just reheat in the oven or microwave for a few minutes when you're ready to assemble the bowl.
Sesame sauce keeps well up to a week, refrigerated. If it thickens up, stir in a tablespoon or so of cold water to loosen it.