In a small bowl or measuring cup, whisk together soy sauce, ketchup, brown sugar, rice vinegar, toasted sesame oil, and cornstarch.
Add vegetable oil to a skillet. Add the minced garlic and the white parts of the green onions *before* turning on the heat. (This will help to prevent the garlic from burning.)
Place the pan over medium heat. Once it begins to sizzle, sauté for 2-3 minutes, or until the whites of the green onions are turning translucent and the mixture is very fragrant.
Give the sauce mixture another whisk to redistribute the cornstarch, then pour the mixture into the pan with the scallions and garlic. It should begin to simmer almost immediately. Stir constantly for 1-2 minutes, and the sauce should thicken up.
Add the crispy tofu, the green parts of the scallions, and the toasted sesame seeds. Toss to coat, and serve immediately.