Preheat oven to 350°F; line two baking sheets with parchment paper. In a medium bowl, mix 4 tablespoons unsalted butter, 3 tablespoons sugar, 2 tablespoons all-purpose flour, 1 teaspoon vanilla extract, and 1/4 teaspoon salt until smooth; mix in 2/3 cup rolled oats.
Drop firmly packed teaspoons of dough, 2 inches apart, onto baking sheets. Bake until golden, rotating sheets halfway through, 12 to 15 minutes. Cool on sheet 1 minute with a very thin metal spatula, transfer to paper towels to cool completely.