Melt the butter in a medium saucepan over medium heat. Add the finely diced shallot and cook, stirring occasionally, for 2-3 minutes until softened and translucent. Add the minced garlic and cook for another 30 seconds until fragrant. Now, pour in the rolled oats and stir them into the butter mixture. Let them toast gently for about 2 minutes, stirring constantly, until they smell nutty and are slightly golden.
Slowly pour in the broth, stirring to combine. Add the salt and pepper. Bring the mixture to a gentle boil, then immediately reduce the heat to low. Let the oats simmer uncovered, stirring every minute or so, for about 12-15 minutes. The oats should be tender, creamy, and have absorbed most of the liquid. If they look too thick before they are done, add a splash more broth or water.
Once the oatmeal is cooked, stir in the handful of spinach. Stir for about a minute until the spinach is completely wilted into the hot oats. Next, add the shredded cheddar cheese and stir until it melts completely into a rich, cohesive mixture. Taste and adjust the seasoning with more salt or pepper if needed. Divide the oatmeal between two serving bowls.
While the oatmeal is finishing, heat a small non-stick skillet over medium heat with a tablespoon of butter. Crack the eggs into the hot skillet, and allow them to cook to your desired doneness. Sprinkle on any desired toppings, such as sliced green onions, red pepper flakes, crispy bacon bits, sautéed mushrooms, or cherry tomatoes. Serve with the oatmeal.