Cut tofu into cubes or slices. Place on a clean tea towel or paper towel to blot off excess liquid while you prepare your remaining ingredients.
In a small bowl, whisk together sesame seeds, brown sugar, soy sauce, vegan oyster sauce, gochugaru, and 1/2 cup water. Set aside.
In a skillet or wok, heat at least 1/2" vegetable oil to 375-400°F. (I highly recommend using a thermometer; if the oil isn't hot enough, the tofu will not brown or crisp, and it will retain more oil.)
Fry tofu for 3-4 minutes per side, or until golden brown and crisp. Remove to a wire rack or a paper towel-lined plate. Work in 2-3 small batches to keep the oil at a consistent temperature. Also, be extremely careful when lowering tofu into the oil! Tofu contains a lot of moisture and tends to splatter a bit. This is why it's important to press out the excess liquid first.
Set fried tofu aside and heat sesame oil in a large skillet or wok over medium-high heat. (I usually pour the used frying oil out into a glass jar, wipe down my wok with a paper towel, and stir-fry in that to avoid using an extra pan.)
Stir-fry onion, garlic, red bell pepper, and the white parts of the green onions for 3-5 minutes, until the mixture starts to brown.
Add in fried tofu, then pour in the sauce ingredients, stirring constantly. The sauce should immediately start to bubble and thicken. Depending on how saucy you want the dish, you can add an extra 1/4 - 1/2 cup water and continue stirring until thickened.