Three hours before dinner, gather two cohorts in the kitchen. One of you makes sofrito: In a 4-qt. saucepan, cook peppers and onions in oil over medium heat until translucent, 8 to 10 minutes. Add tomato purée, season with salt and pepper, and simmer over low heat, un- covered, into a thick sauce, 5 minutes. Heat a grill to medium (350° to 450°), with a heaping chimney of coals if using charcoal.
Meanwhile, the other two prep vegetables for paella. For artichokes: Bring a large pot of well-salted water to a boil. Snap off outer leaves down to tender yellowish ones, and trim fibrous outer layer of stem. Cut off spiny top of each artichoke, then halve them lengthwise and scoop out any fuzzy chokes. Simmer artichokes until just tender, 12 to 15 minutes, and cool in a bowl of ice water.
For asparagus: Bring another pot of well-salted water to a boil. Blanch asparagus until tender-crisp, about 2 minutes, then cool in another bowl of ice water. If using English peas, blanch until just tender, 2 to 3 minutes, and add to ice water. Lift asparagus onto a plate and drain peas in a colander.
About 1 1/2 hours before dinner, form a caravan to the grill with prepped ingredients, paella pan, oven mitts, and a long-handled wooden spoon. Take turns cooking between sips of wine.
Heat pan, then drizzle in olive oil and swirl to coat. Add mushrooms, season with salt, and cook until browned, 5 minutes. Stir in sofrito, smoked paprika, and saffron.
Add broth (11 cups if using bomba, 7 cups if using any other rice), 11/2 tsp. salt, and 3/4 tsp. pepper and bring to a lively simmer. Taste and add more salt and pepper if need- ed (especially if using more broth); broth must be on the salty side to properly season the rice. Sprinkle rice evenly over broth and simmer (covered if on gas), until grains start to become visible, 5 to 10 minutes.
Arrange artichokes, asparagus, and peppers over rice. Cook paella (still covered if on gas) until rice has ab- sorbed all of liquid and smells slightly toasted; then cook for a couple of minutes more to form a soccarat, or bottom crust (insert a spoon into the rice—if it feels a little bumpy, it has a crust).
Remove paella from heat. Add blanched or thawed frozen peas, cover with foil, and let rest 5 minutes.
Escort paella from grill to center of the table. Lay down lemon wedges and olive oil, for drizzling.
*Or soak 2 oz. dried morels in warm water until soft, 20 to 30 minutes. Squeeze gently to release grit, then trim stems and slice in half. Use liquid (carefully pour off grit in bowl) as part of cooking broth.