Begin by marinating the portobello. Cut the mushrooms into ½" strips and set aside. Mix together the ingredients for the marinade in a flat, shallow dish. Place the portobello strips in the marinade, submerging them completely. Allow mushrooms to marinate for at least 30 minutes.
While the mushrooms marinate, heat one tablespoon of olive oil in a medium sauté pan placed over medium heat. Add sliced onions and reduce heat to low. Saute for 15-20 minutes, stirring frequently, until the onions are golden and caramelized. Remove the onions from heat and set aside.
In a small bowl mix together the ingredients for the basil aioli. Place in the refrigerator until ready to use.
Transfer the portobellos, along with the marinade, into the hot sauté pan. Turn the heat up to medium-high and bring the liquid to a simmer. Heat until the liquid is absorbed and mushrooms are browned and have reduced in size by about 50%.
To assemble the sliders, spread the basil aioli on the bottom side of all four buns. Evenly distribute greens, portobello, and caramelized onions. Top with the other half of the bun and serve immediately.