Heat oven to 425 degrees F. Drain and dice tofu. In a medium bowl add tofu and toss with cornstarch. Spread on 1/2 of the baking tray with silpat.
Peel and cube eggplant into 1 inch pieces. In a large bowl paced diced eggplant and toss with oil. Spread on 1/2 of the baking tray with silpat.
Roast tofu and eggplant in oven for 20 minutes to start. After 20 minutes, use your thinnest spatula to gently separate the tofu from the pan and flip to crisp and brown on the other side, about another 10 minutes. Do the same with the eggplant. At 30 minutes, the tofu should be crisp and browned and the eggplant should be roasted and tender. If needed, cook it for 5 more minutes.
While tofu and eggplant roasts, prepare the sauce. Heat a large, heavy skillet over medium heat and add butter. Once butter melts, add shallots, garlic, and ginger. Reduce heat slightly and cook, stirring here and there, until everything is tender, about 11 to 14 minutes. Add the soy sauce, sugar, and black pepper and cook, simmering, for 3 minutes more.
Add roasted tofu and eggplant to pan and stir to coat with sauce; cook for one to two minutes minute together. Serve over or with rice; add extra heat as needed.