Heat the olive oil in a large skillet over medium heat. Add the sliced onions and cook, stirring frequently, until soft and light golden brown, about 15 minutes.
Stir in the balsamic vinegar, brown sugar, fresh thyme, salt, and pepper. Cook for an additional 5 minutes to let the flavors fuse, then remove the skillet from the heat. Divide the onions, reserving 1/3 of them in a small bowl for the topping.
Preheat the oven to 375°F (190°C).
In a food processor, combine the tofu, nutritional yeast, Dijon mustard, coconut amino, plant-based milk, arrowroot, minced garlic, and salt and pepper. Blend until smooth and creamy.
Fold in 2/3 of the onions into the tofu mixture. Reserve the rest of the onions to put on top.
Take out crust from refridgerator. Spread the tofu mixture evenly over the bottom of the tart crust. Spoon the reserved 1/2 of the onions across the top, spreading them out evenly. Press the top layer of onions lightly down into the surface of the tofu mixture to insulate from direct heat.
Loosely place aluminum foil over the tart for a tent that slightly arches at the top so it doesn't stick to the onions. Place it in the preheated oven and bake for 20 minutes.
Take aluminum foil off and then bake for another 20 -25 minutes until top is golden brown.
Allow the tart to cool for a few minutes before slicing.