In a small pot, bring 3/4 cup of water to a boil. Stir in the oats and a pinch of salt. Reduce heat to low and cook for about 5 minutes, stirring occasionally, until the oats are soft and have absorbed most of the water.
Remove the pot from heat and stir in the shredded cheddar cheese until melted and well combined. Season with salt and pepper to taste.
While the oats are cooking, heat half a teaspoon of coconut oil in a small skillet over medium heat. Add the diced red pepper and chopped onions, sautéing until they are soft and slightly caramelized, about 3 minutes. Set aside.
In the same skillet, add the remaining coconut oil and crack the egg into the skillet. Fry the egg to your desired doneness, seasoning with a pinch of salt and pepper.
Spoon the cooked oatmeal into a bowl. Top with the sautéed vegetables and fried egg. Garnish with chopped walnuts and sliced green onions (optional) alongside any other toppings you desire.