Place a steamer insert into a saucepan and fill with water to just below the bottom of the steamer. Bring water to a boil. Add cabbage, carrots, and broccoli, cover, and steam until tender, 2 to 6 minutes.
While vegetables are steaming, heat 1 tablespoon oil in a large skillet over medium heat. Add onion; cook and stir 2 to 3 minutes. Add mushrooms and cook about 5 minutes more. Mix steamed vegetables into the skillet; add spinach, aminos, and hoisin sauce. Cook filling until spinach is limp, about 1 minute.
Fill a bowl the size of a rice paper sheet halfway with water. Soak 1 sheet at a time until just softened. Place rice paper sheet on a clean work surface. Spoon 2 to 4 tablespoons of filling onto the middle of the sheet and roll the bottom up half-way. Fold sides in, burrito-style, and continue rolling up snug. Place filled roll on a plate. Repeat with remaining rice paper sheets and filling. Set rolls in the refrigerator for 5 minutes; flip over and refrigerate until dry, about 5 minutes more.
Preheat the oven to 200 degrees F (95 degrees C).
Heat remaining 1 tablespoon oil in a skillet over medium heat. Fry rolls on each side, turning periodically to prevent over-browning, until the rolls are a delicate golden color all over, 3 to 5 minutes. Place fried rolls on a baking sheet in the oven to keep hot until ready to serve.