In a small bowl, toss 1 pint strawberries, rinsed, hulled, and thinly sliced, with 1 to 2 tablespoons firmly packed dark-brown sugar. Let sit at least 5 minutes to bring out the juices.
In a medium saucepan over medium heat, warm 1 3/4 cups water, 1 3/4 cups milk, and 1/4 teaspoon salt. Stir in 2 cups quick-cooking oats; cook, stirring occasionally, until thick and creamy, 5 to 6 minutes.
Ladle oatmeal into bowls. Top each serving with 1 tablespoon sour cream and some of the strawberries; sprinkle with brown sugar.