Soak sundried tomatoes in boiling water for 10-15 minutes, until they become soft again. Chop sundried tomatoes finely, but still allowing for their texture to come through.
Remove mushroom stems and set aside. Clean mushrooms well (this depends on how you like to do it. Some use a pastry brush; others peel the 'shrooms. I like to wipe them clean with a wet paper towel. But I feel strongly against washing them - it ruins their texture). Chop stems finely, leave mushroom caps whole.
Drain tofu and chop finely.
Chop onion finely. Sauté until it becomes translucent, about five minutes. Add chopped mushroom stems and sauté another five minutes, until they start to give out juices. Add sundried tomatoes and tofu. Let cook for about 10 minutes on medium heat, letting the juices intermingle.
Chop garlic and add. Add all spices, herbs and balsamic vinegar. Cook on medium-high heat for 10-15 minutes, stirring often, until mixture dries out a bit and the tofu colours.
Preheat oven to 375F.
Line a baking sheet with aluminum foil. Stuff mushrooms and arrange them on sheet. Top with grated vegan cheese. Drizzle with a bit of olive oil. Bake for 8-10 minutes.
Right before serving, broil/grill mushrooms at 400F for three minutes, to heat up the cheese. Sprinkle some chopped basil on top right before serving.