Heat the olive oil in a large, straight-sided saute pan over medium-high heat. Add the onions and saute until soft, about 8 minutes. Add the oats and toast, stirring often, until they smell nutty, about 2 minutes. Season with salt and pepper.
Add 2 1/4 cups of the chicken stock and 2 1/4 cups water and bring to a simmer. Cook, stirring often, until the oatmeal has absorbed most of the liquid. Add more of the chicken stock or water as needed and continue to cook until the oatmeal is tender, about 25 minutes total.
Stir in the butter along with half of the basil, mint and cheese. Divide the oatmeal among serving bowls and top each with some of the remaining basil, mint and cheese and a drizzle of olive oil.