Heat 2 tablespoons of the canola oil in a large non-stick skillet over medium-high until shimmering. Add half of the tofu in a single layer (do not overcrowd). Cook until golden and crisp on the first side, about 3–4 minutes, then turn and cook the other side until golden. Transfer the cooked tofu to a plate lined with a paper towel.
Add the remaining tofu to the same skillet (no additional oil needed) and cook the second batch the same way, 3–4 minutes per side, until golden. Transfer to the plate with the first batch to drain.
Wipe the skillet lightly if needed, then add 2 tablespoons of the canola oil and heat over medium-high. Add the mushrooms, toss to coat in the oil, then add half of the sliced green onions. Cook, stirring occasionally, until the mushrooms are softened and starting to brown, about 5 minutes. Transfer the mushrooms to the plate with the tofu.
Add the last 2 tablespoons of canola oil to the same skillet and heat over medium-high. Add the cold cooked rice, break up any clumps with a spatula, spread the rice into an even layer, and cook without stirring for about 2 minutes to let the bottom get slightly crisp.
Add the chopped kimchi, the reserved 1/4 cup kimchi juice, the 3 tablespoons soy sauce, and the 1 tablespoon toasted sesame oil to the rice. Toss or stir to combine and cook for 2–3 more minutes, until the rice is evenly coated and heated through.
While the rice is warming, melt the 1 tablespoon butter in a separate non-stick skillet over medium heat. Whisk the 4 eggs in a bowl, pour into the hot skillet, and cook while stirring gently until soft curds form (scrambled but still moist). Remove from heat.
Add the scrambled eggs, the fried tofu, and the cooked mushrooms to the rice. Gently toss everything together in the large skillet and cook for 1–2 minutes more to warm the tofu and mushrooms through.
Taste and, if desired, adjust the flavor using a little more soy sauce or sesame oil. Sprinkle the remaining sliced green onions over the top and serve immediately.