Green Chilaquiles With Eggs

fresh tomatillos, husked and rinsed

to 4 jalapeño or serrano chiles, stemmed, to taste (seeded for a milder salsa)

cilantro sprigs, plus 1/3 cup chopped cilantro

olive oil or grapeseed oil

small white onion, quartered and thinly sliced (about 1 cup sliced onion)

Salt

large garlic cloves, minced

chicken or vegetable broth

large eggs

thick corn tortillas, cut into wedges and deep-fried or microwaved until crisp (see note), or 1/2-pound thick tortilla chips from a Mexican grocery

crumbled queso fresco or feta