Preheat the oven to 375 degrees Fahrenheit. Spread bread pieces on a baking sheet in a single layer and bake for 20-30 minutes until lightly toasted. Set aside.
While the bread is toasting, heat the vegan butter in a deep skillet or enamel dutch oven. Add the sausage and sauté until golden brown, about 5 minutes.
Add the onion, leek, carrots, celery, and mushrooms. Cook for 7-9 minutes or until onion is softened and translucent.
Stir in the white wine, rosemary, sage, thyme, salt and pepper. Cook for a couple of minutes until the wine has evaporated. Transfer to a 9" x 11" baking dish and set aside.
Pour in the bread cubes and stir to combine.
In a medium mixing bowl, add the broth and Just Egg. Stir well with a whisk. Pour over the bread and vegetable mixture in the baking dish. Stir gently.
Cover with aluminum foil and bake for 35 minutes. Uncover and bake and additional 10-15 minutes or until top is browned.