Make the aïoli
Grind the saffron threads in a mortar and pestle or spice grinder until powdery; you should have about 1/8 tsp.
Blend the egg, yolk, garlic, vinegar, mustard, and 1/4 tsp. salt in a blender or food processor until well combined, about 10 seconds. With the machine running, slowly drizzle in the vegetable and olive oils until thick and emulsified, about 1 minute. Blend in a large pinch of the saffron (save the remainder for the fideos). Season to taste with salt. The aïoli will become more yellow in color as the saffron blooms. Stir before using.
Make the sofrito
In a 12-inch ovenproof paella pan or heavy-duty skillet, heat the oil over medium-low heat. Add the onion and bell pepper and cook, stirring occasionally and then smoothing the mixture back into an even layer in the pan, until the onions are golden and soft, 5 to 8 minutes. Add the tomato and continue to cook, stirring and smoothing the mixture, until the vegetables are a shade darker, about 2 minutes. Add the garlic and tomato paste and cook, stirring occasionally, until the oil begins to separate and the mixture is yet another shade darker, about 2 minutes. Stir in the paprika and remove from the heat. Season to taste with salt and transfer to a bowl to cool. Clean the pan.
Make the fideos
Heat the vegetable oil in a 2-quart saucepan over medium heat to 325°F. While the oil heats, set a sieve over a medium pot.
Add the fideos to the oil and fry, gently stirring with a wooden spoon, just until they turn golden brown, about 1-1/2 minutes. Immediately pour the fideos and oil into the sieve; the fideos will continue to cook and become dark golden brown as they drain and cool. Set aside. Reserve the oil for frying more fideos or another use.
Position an oven rack in the lower third of the oven and heat the oven to 500°F.
Heat 1 Tbs. of the olive oil in the cleaned paella pan or skillet over medium heat until shimmering hot. Add the chicken and cook, stirring occasionally, until no longer raw on all sides, 2 to 3 minutes; transfer to a plate. Add the remaining 1 tsp. olive oil and the garlic and cook, stirring, until fragrant and light golden, about 30 seconds. Add the reserved sofrito and cook, stirring occasionally, until heated through, about 1 minute. Add the chorizo and cook, stirring, until heated through, about 1 minute. Add the chicken, any juice from the plate, and the wine, and cook, stirring and scraping the browned bits from the bottom of the pan, until the liquid is almost evaporated, about 1 minute. Add the fideos and then sprinkle the remaining saffron and 1/2 tsp. salt over them. Slowly pour in the stock and then spread the mixture into an even layer, if necessary; the stock should just barely cover the fideos. (It’s OK if some fideos tips stick out of the stock). Turn the heat up to medium-high and cook, undisturbed, until boiling in the center, about 5 minutes. Transfer to the oven and bake until most of the stock is absorbed by the fideos, about 10 minutes.
Push the mussels into the fideos at even intervals. Bake until the mussels open, 3 to 5 minutes. Transfer the mussels to a plate and cover with foil to keep warm.
If the mussels have released a lot of liquid, put the pan over medium-low heat, stirring once or twice, until the liquid is absorbed, about 5 minutes.
Gently fold in the fava beans, if using, and let stand at room temperature for 5 minutes.
Gently fold in 2 Tbs. of the aïoli and the chopped parsley. Season to taste with salt, if necessary. Garnish with the mussels and parsley leaves, and serve.
The aïoli can be made up to 5 days ahead; refrigerate in an airtight container and bring to room temperature before using.
The sofrito can be made up to 3 days ahead; cool completely and store in an airtight container in the refrigerator.
You can fry the fideos up to 2 days ahead; store in an airtight container at room temperature.