Combine the broth, rice, half of the garlic, and salt in a small saucepan. Bring to a boil over medium-high heat, then cover and reduce the heat to low. Cook until the rice is tender and the broth is absorbed, about 20 minutes. Remove from the heat and let sit, covered, for 10 minutes to finish steaming. Stir in the cilantro and fluff with a fork.

Heat the oil in a Dutch oven over medium heat until shimmering. Add the onion and cook until it just begins to brown, about 5 minutes. Stir in the tomato paste, chipotle, cumin, oregano, remaining garlic, and salt. Cook until the spices are fragrant, about 1 minute.

Add the chard and half of the remaining broth to the pot. Cover and simmer until the chard is tender, about 15 minutes.

Coarsely mash half of the black beans with the final portion of broth in a medium bowl using a potato masher. Stir the mashed beans into the Dutch oven and cook, stirring constantly, until the liquid is nearly evaporated and the mixture is thick, about 3 minutes. Off the heat, stir in the lime juice and the remaining whole beans.

Position an oven rack 6 inches from the broiler element and heat the broiler. Wrap the tortillas in a damp dish towel and microwave until warm and pliable, about 1 minute. Lay the tortillas on a clean surface and mound the rice, chard-bean mixture, and a portion of the Monterey Jack across the center of each, close to the bottom edge.

Fold the sides of the tortilla over the filling, then fold up the bottom edge, pulling back firmly to tighten it around the filling as you roll into a tight cylinder. Place the burritos seam-side down on an aluminum foil–lined baking sheet and sprinkle the remaining cheese over the top.

Broil until the cheese on top is melted and starting to brown, 3 to 5 minutes. Serve hot.