In a large saucepan with a lid, bring 1 quart water to a boil. Add chopped onion, 3 garlic cloves, carrot, celery, spices, and chile pepper, stirring to combine. Add pork, cover, and simmer 30 to 45 minutes, until pork is tender. Season with salt. Remove pork from stock and set aside. Strain stock, discard seasonings, and set remaining stock aside.
Meanwhile, in a small saucepan over medium-high heat, cover tomatillos and jalapeños with 1 cup water. Simmer until tomatillos begin to change color, from bright green to dull brown-green, about 8 to 10 minutes. Carefully transfer hot tomatillos, jalapeños, and any remaining liquid to a blender or food processor. Add the quartered ¼ white onion and garlic and blend until smooth (alternatively, blend in the stockpot using a stick or immersion blender). Set aside.
Heat oven to 350ºF. In a deep, heavy-bottomed frying pan, heat oil or lard over high heat until smoking. Add pork and fry, turning as necessary, until well browned, about 10 minutes. Lower heat, add tomatillo-jalapeño mixture, and stir. Add verdolagas leaves and ¾ cup reserved stock. Cover and cook over low heat about 20 minutes. Season with salt.
Spoon mixture into 4-ounce cazuelitas or small ramekins and top with cheese. Arrange ramekins on a rimmed baking sheet and bake until cheese bubbles and begins to brown, about 20 to 25 minutes. Serve with fresh corn tortillas.