In a large skillet over medium-high heat, heat 3 tbsp. oil. Add finely chopped onion, cumin, 1 ½ tsp. salt, and 1 tsp. black pepper. Cook, stirring occasionally, until onion has softened, about 5 minutes. Add beans and 1 c. broth and cook, stirring occasionally, until most liquid has reduced, 5 to 7 minutes. Continue to cook, using a potato masher or fork to smash beans, until smooth, 3 to 5 minutes. Remove from heat and stir in lime juice.

Meanwhile, in a medium pot over medium heat, heat remaining 2 tbsp. oil. Add sliced onion and remaining 1 ½ tsp. salt and 1 tsp. black pepper. Cover and cook, stirring occasionally, until onions have softened slightly, 3 to 5 minutes. Add garlic and jalapeño, if using, and cook, stirring, until fragrant, about 1 minute. Stir in green chiles and remaining 1 c. broth. Increase heat to medium-high and cook, stirring occasionally, until reduced by half, 5-7 minutes. Stir in chicken until warmed through. Remove from heat and stir in cilantro.

Top tostadas with bean mixture, chicken mixture, romaine, avocado, and queso fresco. Garnish with more cilantro. Serve with lime wedges alongside.