Heat vegetable oil in a large pot over medium-high heat; saute onion for 5 minutes. Add mushrooms and garlic; cook, stirring regularly, for another 5 minutes. Cover pot between stirs. Add spinach, red wine, balsamic vinegar, and liquid smoke. Stir briefly, cover the pot, and remove from heat. Let cool.
Preheat the oven to 350 degrees F (175 degrees C). Arrange pie crust in a 9- or 10-inch pie plate.
Combine tofu, arrowroot, nutritional yeast, olive oil, mustard, and agar powder in a food processor; blend into a uniformly thick paste. Mix into the mushroom mixture to form a more liquid paste. Pour and scrape mixture into the pie crust.
Bake in the preheated oven until crust is golden and center is set, about 45 minutes. Cool quiche completely before slicing.