Heat 1 tablespoon oil in a large saucepan over medium heat. Add shallot and cook, stirring occasionally, until softened, 1 to 2 minutes. Add oats and stir for 1 minute. Add 4 cups water and ¼ teaspoon each salt and pepper. Bring to a boil, then reduce heat to a simmer. Cook, stirring often, until creamy, 10 to 12 minutes.
Meanwhile, heat the remaining 1 tablespoon oil in a large skillet over medium-high heat. Add collards along with the remaining ½ cup water and ¼ teaspoon each salt and pepper. Cook, stirring occasionally, until tender, 5 to 7 minutes. Remove from heat and stir in vinegar.
Stir cheese and salsa into the oatmeal. Serve with the collards, eggs and more salsa, if desired.
