Combine oats, water or milk, and salt in a medium saucepan over medium heat.

Bring to a boil, then reduce heat to low and simmer for 5-7 minutes, stirring occasionally, until oats are tender and creamy.

Chop the drained peaches into bite-sized pieces and stir into the cooked oatmeal along with reserved syrup, cinnamon, and vanilla extract.

Let stand for 2 minutes to allow peaches to warm through, then taste and add brown sugar if desired.

Serve hot in bowls, optionally topped with additional cinnamon or a pat of butter.