In a medium skillet, heat the oil over medium heat. Add the mushrooms and cook, stirring often, until the water has been released, about 5 minutes.
Transfer mushrooms to a food processor. Add the spinach, beans, flax eggs, walnuts, almond flour, tomato paste, nutritional yeast, Italian seasoning, salt, lemon zest, black pepper, onion powder, and garlic powder. Pulse until well combined yet retains structure. Transfer to a bowl. Cover and chill for 30 minutes.
Preheat oven to 400°F. Line a large rimmed baking pan with parchment paper. Shape bean mixture into 12 meatballs. Roll meatballs in almond flour. Place meatballs on the baking pan. Bake until browned and heated through, 25 to 30 minutes.
Meanwhile, make the pesto. In a large skillet, cook the noodles until just tender, 4 to 5 minutes, tossing occasionally. Pour the pesto over the noodles and toss to coat. Serve the meatballs with the pesto noodles.