Heat 2 tbsp olive or avocado oil over medium heat. Toss shiitakes in oil with some salt and pepper and cook undisturbed for 5 minutes. Shake mushrooms around the pan and continue to cook about 5 more minutes until starting to crisp and caramelize. Remove half of the mushrooms to a plate.
To the same pan, saute garlic, onion, red pepper, and paprika for 2-3 minutes until fragrant and the pepper softens slightly. Slowly pour in wine and scrape the bottom of the pan with a wooden spoon. Continue to cook about 3-4 minutes until wine reduces by half.
Stir in canned tomatoes, edamame, saffron threads, and rice. Pour broth into the pan, season with salt and pepper, and gently move the components around to distribute evenly. After this, DO NOT TOUCH THE PAELLA! Cover the pan and bring liquid to a gentle simmer. Cook, covered, for 15 minutes.
Remove the lid from the pan. WITHOUT STIRRING, sprinkle on white beans and wedge olives across the surface. Continue to cook for 10 minutes uncovered. Remove pan from the heat and replace the lid, then toss a towel over the lid and let the paella sit for another 8-10 minutes.
Just before serving, pour lemon juice into the paella and garnish with chopped parsley. Serve warm, alone or with crusty bread.