5 tablespoons extra-virgin olive oil (preferably Spanish), divided
1 pound uncooked large shrimp, peeled, deveined, butterflied
1 pound skinless boneless chicken thighs, cut into 1-inch cubes
1/2 pound oyster mushrooms, sliced
1 cup chopped onion
4 garlic cloves, chopped
1 1/2 cups paella rice (short-grain white rice)
1 15-ounce can diced tomatoes in juice
1 dried California chile or New Mexico chile, stemmed, seeded, finely ground in spice mill
3/4 teaspoon smoked paprika
1/2 teaspoon saffron threads, crumbled
2 cups low-salt chicken broth, divided
1 cup mushroom broth