Combine peaches, granulated sugar, lemon juice, cinnamon, and water in a medium saucepan over medium heat. Stir to coat the fruit and bring the mixture to an active simmer (bubbles around the edges, 1–2 minutes).

Cook for 10–15 minutes, stirring occasionally, until peaches are very soft and have released their juices and the liquid has thickened to a light syrup. Look for an amber sheen and visible peach chunks suspended in the syrup.

Remove from heat and stir in vanilla extract. The sauce should smell warm and taste slightly concentrated (about 30–60 seconds of mixing).

For a smoother compote, mash some of the peaches with the back of a spoon or use an immersion blender briefly. Stop when the texture is still chunky but more uniform (about 10–20 seconds).

Serve warm over ice cream, pancakes, waffles, yogurt, cheesecake, or pound cake. The compote should spoon with a glossy drip and amber syrup streaks on the surface.

Store covered in the refrigerator for up to 1 week and reheat gently to serve warm. Warm until just pourable, not boiling, so the syrup stays glossy (about 2–4 minutes).