In a medium saucepan, heat the olive oil over medium-low heat. Add the shallots and cook briefly until translucent, then add the mushrooms. Cook for several minutes, stirring periodically, until the mushrooms have softened.
Add 2 cups of water and bring to a boil. Once the liquid is boiling, add the oats, salt, and pepper, then reduce to a simmer. Cook for 10-20 minutes, stirring periodically, until the desired texture and consistency is reached.
While the oatmeal is cooking, heat a large pot of water with the vinegar to approximately 180 degrees F (clip a thermometer to the side of the pot if you have one). Crack the eggs into separate, small measuring cups and then gently drop the eggs into the water, one at a time. Cook for 4-6 minutes (depending on how firm you like your egg), then use a slotted spoon to remove the egg. Briefly drain on a paper towel.
Spoon the oatmeal into two bowls. Add a poached egg, sliced tomatoes and basil to each serving. Top with a generous amount of freshly grated parmesan cheese and a sprinkle of ground black pepper.
Notes
You can swap in steel cut oats for the rolled oats, just note that they take longer to cook.Fried or sunny side eggs would also work well with this recipe.