Preheat oven to 375° F. Grease or paper line a standard muffin pan with 12 wells.
In a small bowl, toss strawberries with balsamic vinegar and a spoonful of the brown sugar. Set aside, stirring once or twice while you prepare the other ingredients.
In a large bowl, whisk together remaining brown sugar, sour cream, egg, vanilla, and oats. Mix in melted butter and set aside.
In a medium bowl, whisk together flour, baking powder, baking soda, and salt.
Add dry ingredients, along with the strawberries and their juices, to the wet ingredients. Fold ingredients together with a silicone spatula until just combined, no more than 10 to 15 strokes, making sure you scrape up from the bottom of the bowl for even mixing.
Divide batter evenly between the prepared muffin tin wells.
Bake until the tops of the muffins spring back when lightly pressed or a toothpick comes out clean, about 15 to 20 minutes.