Preheat oven to 375 degrees Fahrenheit. In a medium bowl, combine brown sugar, butter, molasses, and vanilla.
Lightly scoop flour into measuring cup to measure 1/3 cup. Stir flour, oats, ginger, and salt into sugar mixture.
Line a baking sheet with a silicon baking mat or parchment paper. Divide dough into 6 portions. Place 3 or 4 portions on prepared baking sheet, leaving space for cookies to spread to 5 to 6 inches in diameter. (Use second baking sheet or make in two batches if needed.)
Flatten dough into 3-inch circles. Bake 8 to 10 minutes, until edges are dark brown. Cookies should bubble and spread during baking.
To shape cookies: cool cookies on the pan for 1 to 2 minutes. Slide a wide spatula under warm cookies, and place over an inverted custard cup or 2- to 3-inch diameter heat-proof container. Gently press cookie around cup. Cool. Repeat with remaining cookies. If cookies crisp before shaping, return to oven briefly until softened.
Fill cookie cups with scoops of dulce de leche ice cream and serve.