I was recently gifted a bottle of goat dulce de leche (with white wine!) and after gorging myself on the sweet treat, I decided to make proper use of it and incorporate it into my baking. The trick is to balance the sweetness of the dulce de leche with a cookie doesn't have too much sugar and generally has a neutral base. I revisited the madeleine pan cookies and made dulce de leche cookie sandwiches with hazelnut shortbread. The shortbread is not only subtle but the hazelnut added an almost-savoury feel to the cookie. In other words, it was a perfect pairing! I think another combination that could work would be a dark chocolate wafer or espresso cookie - using cocoa powder or espresso could add that dense bitterness that would offset the sweetness of the filling quite well.
Photographic evidence for your pleasure:
(goat dulce de leche with white wine)
(dulce de leche...is...very...fluid - use less, MUCH less to sandwich than you think you need - the day after a few of my cookies had become separated simply from sliding off the filling)