To make the dressing, combine lime juice, sugar, garlic, and chile in a small bowl. Whisk in the fish sauce, tahini, and water. Taste and add more lime, sugar, or chile as needed. Sauce will keep well covered in the fridge for several days.
Cook the rice noodles according to instructions. Drain, rinse well with cold water, and drain again.
Meanwhile, heat a small skillet over medium heat. Add the olive oil, and then the mushrooms and 2 to 3 tablespoons of the prepared dressing. Cook for 5 minutes, or just until the mushrooms are soft.
To serve, toss cooked noodles with several tablespoons dressing. Toss the spinach with a tablespoon of dressing, and set atop the noodles. Add carrots, mint, cilantro, and green onion to taste. Top with warm mushrooms, another drizzle of dressing, and a squeeze of lime.