How to Make It
Heat 2 tablespoons oil in a 15-inch carbon steel paella pan over high until shimmering. Add squash and mushrooms; cook, undisturbed, until browned on 1 side, about 3 minutes. Continue cooking, stirring occasionally, until browned in all over, about 3 minutes. Transfer to a small bowl; set aside.
Reduce heat to medium-low; add onion, bell pepper, 1/2 teaspoon salt, and remaining 3 tablespoons oil. Cook, stirring often, until onion darkens around edges, about 15 minutes, stirring in garlic during final 1 minute of cook time.
Stir diced tomatoes and tomato paste into mixture, scraping up browned bits on bottom of pan with a wooden spoon. Cook, stirring often, until liquid in pan evaporates and tomato paste sizzles, 3 to 5 minutes. Stir in vegetable stock, saffron, and remaining 1 teaspoon salt. Increase heat to high, and bring to a boil. Stir in chard and squash mixture. Sprinkle rice over mixture, stirring gently to disperse evenly.
Return mixture to a boil over medium-high. Boil until rice grains are just above top of stock, about 10 minutes.
Reduce heat to medium-low; simmer, repositioning pan over heat every few minutes by turning pan about 90 degrees to redistribute heat, until rice is tender, has absorbed all liquid, and begins to brown on bottom, about 20 minutes. (Use a small spoon to gently lift rice from bottom of pan to check for brownness on crust.) If the liquid is cooking off quickly and rice is still al dente, add hot water as needed (1/4 cup at a time) until rice is tender.
Remove pan from heat; loosely cover with aluminum foil or a clean kitchen towel, and let stand 10 minutes. Uncover and squeeze juice from lemon half over paella; top with dollops of lemon aioli.
Bomba rice is a short-grain rice that can absorb a large amount of liquid without bursting or becoming mushy. Find it at specialty stores or on amazon.com.