Adjust oven rack to lowest position and heat oven to 450 degrees. Coat a 13- by 9-inch baking pan with 2 tablespoons of oil and set in preheating oven. Meanwhile, season grated potatoes with 1 teaspoon of salt and several grinds of pepper; toss to coat. Remove pan from oven and spread potatoes over pan bottom, pressing to compact. Return to oven and bake until cooked through and golden, about 20 minutes.
Meanwhile, heat a light drizzling of oil, along with garlic and pepper flakes, in a small Dutch oven. When garlic starts to sizzle, add kale; stir to heat through. Add coconut milk; continue to stir to heat through again, lightly seasoning with salt and pepper; stir in nutritional yeast. Remove from heat.
While kale cooks, heat remaining 2 tablespoons of oil in a large skillet over medium-high heat. Add peppers and onions; sauté, seasoning lightly with salt and pepper, until softened, about 5 minutes.
To assemble, remove potatoes from oven. Spread kale evenly over the potatoes and then onions and peppers over the kale. Sprinkle with cheese. Continue to bake until cheese melts and starts to turn golden brown, about 20 minutes. Remove from oven, cool for a few minutes, cut, and serve.