In a medium nonstick skillet over medium-high heat, heat the oil until shimmering. Add the mushrooms and cook, stirring occasionally, until browned and crisp, about 6 minutes. Sprinkle the mushrooms with the paprika and salt, and cook, stirring, for 10 seconds more. Remove from the heat and transfer the mushrooms to a plate.
Cook the oats according to the directions on the package, without adding salt. Once the oats are cooked, stir in the parmesan cheese, remove from the heat and cover to keep warm.
Return the skillet to medium heat. There should still be some oil in the pan, but add 1 teaspoon more, if needed. Crack the eggs into the pan and cook until the whites are set and the yolks are to your desired consistency, flipping them, if desired.
Divide the oatmeal between two bowls. Swirl 1 ½ teaspoons of the pesto into each bowl, then top each with half of the shiitake “bacon” and one of the eggs. Season to taste with salt and pepper, and serve right away.