Fry the tortillas and butternut. Heat the oil to 375 degrees. In batches, fry the tortilla strips and butternut in the hot oil until crisp, about a minute. Drain on paper towels. In a medium bowl toss the butternut strips with the corn starch, shake off excesss. In batches, fry the butternut in the hot oil until crisp, about 4 to 5 minutes. Drain on paper towels.
Make the brothy sauce. In a blender or food processor, combine the drained tomatoes and chipotles. Process until smooth. You should have about 2 ½ cups.
Set a medium (4- to 5-quart) pot or Dutch oven or a large (12-inch) deep skillet over medium heat. Film the bottom with a couple of tablespoons vegetable oil, and, when it’s hot, add the onion, stirring until it is soft and richly browned, about 7 minutes. Add the garlic, stir for a minute, then raise the heat to medium-high. Add the tomato puree and stir nearly constantly until reduced to the consistency of tomato paste, 7 or 8 minutes. Stir in the broth, bring to a simmer, then taste and season with salt (usually about 1 teaspoon).
Finish the chilaquiles. Add the chard to the pot. When the mixture returns to the boil, add the tortilla and butternut strips, stir a couple of times to coat everything evenly, turn off the heat and wait for a minute until the tortilla strips are as soft as you like.
Spoon a portion of chilaquiles onto each of 4 warm deep serving plates (a pasta bowl works perfectly), top with chicken or one of its substitutes, sprinkle with the crumbled goat cheese, drizzle with a little of the thinned Greek yogurt, and strew with cilantro leaves.