Cut tofu into 1/4-in.-thick slices, then gently press dry between doubled paper towels.
Whisk 2 tbsp. grapeseed oil, garlic, sesame oil, and soy sauce in a wide, shallow bowl. Dip tofu slices in marinade to coat both sides. Heat 1 tbsp. grapeseed oil in a frying pan over medium heat; cook tofu until golden and crisp, 15 minutes. Let cool; slice lengthwise into thin strips.
Meanwhile, heat remaining 1 tbsp. grapeseed oil in frying pan over medium-high heat. Add mushrooms and cook, stirring occasionally, until softened, about 5 minutes.
Dip a rice paper wrapper in warm water for a few seconds--just until it's pliable. Lay on plate and let sit 10 to 15 seconds, so the water soaks in.
Layer one-sixth of fillings on the bottom third of the wrapper, starting with spinach.
Lift filled edge up and over fillings and keep rolling it away from you, tucking fillings in firmly but gently and folding in the sides. (Stretching the wrapper as you roll helps.) Repeat with remaining ingredients.