Preheat your oven to 350°F (175°C).
Line a 9x13 inch baking pan with parchment paper, leaving some overhang on the sides.
In a large mixing bowl, cream together the softened unsalted butter and packed brown sugar until light and fluffy.
Add the large egg and vanilla extract to the creamed mixture and mix until well combined.
In another bowl, whisk together the all-purpose flour, baking soda, and salt. Gradually add this dry mixture to the wet ingredients, stirring until just combined.
Gently fold in the quick-cooking oats until they are evenly distributed.
Spread half of the oat mixture into the prepared baking pan, pressing it down evenly.
Spoon the dulce de leche over the bottom layer and spread it evenly.
Sprinkle the remaining oat mixture over the dulce de leche layer and gently press it down.
Bake for about 25-30 minutes, or until the top is golden brown and set.
Let the bars cool completely in the pan before lifting them out and cutting into squares.