Preheat the oven to 350 F. Wash and dry the mushrooms, then pop out and discard the stems.
In a pan over medium heat, add two tablespoons of butter then toss in the pitted dates and saute until tender.
Add the remaining butter and green onions, garlic, and rosemary; stir until fragrant. Pour a splash of wine into the pan to deglaze it. Let the wine evaporate completely, while scraping up the browned bits, then remove from the stove.
Next, add two tablespoons of vegetable oil into a small pan over medium high heat. Place the mushrooms in the pan, top down, and let brown for one to two minutes, then flip the mushrooms stem side down and repeat. Remove the mushrooms to a plate and let cool.
Cut the brie into cubes about a half in thick - just small enough to fit inside the cavity of each mushroom.
Arrange the slightly cooled mushrooms into a baking dish so the sides are just touching. Press a cube of brie into each mushroom cap, then pour the dates and green onions over the mushrooms.
Bake for 10 to 15 minutes, or until the brie is melted completely. Serve right away warm or chill for a few hours to serve cold. Sprinkle with fresh parsley for garnish.