Preheat the oven to 400°F.
Arrange the portobello caps on a work surface. Remove the stems, and brush broth sides of each cap with olive oil. Season with salt and pepper. Bake the mushrooms (belly-side up) on a parchment-lined baking sheet for 15 minutes, until browned and tender.
In the meantime, warm the butter in a small Dutch oven or sauce pan over medium-low heat. Add the onion and sauté until translucent, about 5 minutes. Add the garlic and the thyme, and cook for another 2 minutes. Deglaze the pan with the wine and simmer for another few minutes, until the alcohol burns off. Set aside to cool.
Arrange the bread on a baking sheet and toast in the oven for 3 minutes, until crusty but not browned. Add the bread to the cooled onion mixture along with the parsley, egg, Parmesan, and ¼ teaspoon salt, and toss to combine.
Remove the mushrooms from the oven. Divide the stuffing between the two caps, pressing down to create a compact mound. Turn the oven temperature down to 350°F, and return the pan to the oven. Bake for 15-20 minutes, until the top of the stuffing is crusty and brown.
Sprinkle with additional parmesan and parsley and serve.