Heat the oven to 325°F with a rack in the lower-middle position. In a large Dutch oven over medium, heat the oil until shimmering. Add the onion and ¼ teaspoon salt; cook, stirring occasionally, until lightly browned, about 7 minutes. Add half of the jalapeños, the garlic and cumin. Cook, stirring, until fragrant, about 30 seconds. Stir in the hominy, pork, broth, 3 cups water and 1 teaspoon each salt and pepper. Bring to a simmer over medium-high, then cover, transfer to the oven and cook for 1 hour.
Remove the pot from the oven. Stir, then return to the oven uncovered. Cook until a skewer inserted into a piece of pork meets no resistance, another 1 to 1½ hours.
Tilt the pot to pool the cooking liquid to one side, then use a wide spoon to skim off and discard as much fat as possible from the surface. Stir in the remaining jalapeños, then cilantro and lime juice. Taste and season with salt and pepper.
