Prepare grill for medium-high heat. Toss 1/2 small white onion, halved lengthwise, keeping root intact, 1/2 head of garlic, unpeeled, halved crosswise, 1 jalapeño, 1 pound husked tomatillos, and 1 tablespoon vegetable oil in a large bowl. Grill vegetables, turning often, until tender and charred, 5-8 minutes.
Transfer to a plate and let cool slightly.
Squeeze garlic cloves into a food processor and pulse with onion, stemmed jalapeño, tomatillos, 1/2 bunch cilantro, leaves and tender stems only, and 1/4 cup fresh lime juice just until a chunky sauce forms; season with kosher salt, freshly ground black pepper, and more lime juice, if desired.
DO AHEAD: Salsa verde can be made 5 days ahead. Cover and chill.
