In a small saucepan, combine the broth with 1 tablespoon of olive oil and a generous pinch of salt. Bring to a boil. Add the oats, return to a boil, then reduce heat to low and cook, stirring occasionally, until thickened, about 13-15 minutes. Remove from heat and cover to keep warm.

In the meantime, in a large sauté pan set over medium-high heat, heat 2 tablespoons of olive oil until shimmering. Add the mushrooms, shallots, dried thyme, if using, salt and pepper to taste and cook, stirring occasionally, until beginning to brown, about 10 minutes.

Divide the oatmeal between serving plates, top with the mushroom and shallot mixture and a fried egg, if using.

Serve immediately.