Heat the oil in a medium skillet. Sauté the onions slowly over medium-low heat, stirring frequently, until nicely browned. Add the mushrooms and cook until they've wilted down.
Combine the onions and mushrooms with the remaining ingredients in the container of a food processor. Process until smoothly pureed, scraping down the sides as needed.
Serve at once with matzo, matzo crackers, or raw veggies. Or store in a tightly lidded container in the fridge until needed, and bring to room temperature before serving.