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.