Preheat oven to 350 degrees, with rack in lowest position. In a saucepan, combine broth, saffron (if using), and paprika; bring to a simmer and keep warm.
In a large skillet (at least 12 inches wide), heat oil over medium. Season chicken with salt and pepper. When oil is very hot, cook chicken, skin side down, until skin is golden, 5 to 7 minutes. Flip, spoon off all but 1 teaspoon fat from skillet, and push chicken to side.
Add chorizo to skillet and cook until browned, about 3 minutes. Add onion and garlic; cook, stirring occasionally, until softened, about 5 minutes. Add tomatoes and cook, stirring frequently.
Add rice; stir to coat thoroughly with tomato-chorizo mixture. Stir in hot broth and 1/2 teaspoon salt (skillet will be very full). Arrange chicken evenly on top, making sure rice is submerged. Bring to a boil and cook, stirring rice occasionally, 8 minutes. Nestle mussels in cooking liquid and cook 2 minutes more. Arrange shrimp on top.
Place skillet in oven and cook until all liquid is absorbed, 10 to 12 minutes. Let paella stand 10 minutes before serving.
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