Place 12 husks in a large bowl, cover with hot water, and weigh down with a plate to keep husks submerged. Soak until soft and pliable, about 2 hours; drain. Peel remaining husks into thin strips to yield 24.
Place poblanos directly over open flame and char, turning with tongs, until blackened on all sides, about 5 minutes. Transfer to a large bowl, cover with plastic wrap, and steam 10 minutes. Using a paper towel, peel skin from peppers. Remove stems and seeds and coarsely chop.
Place a corn husk in your flattened palm. Pinch 1/4 cup dough with other hand and press into center of husk to make a 4-by-3-inch rectangle.
Press 1/4 cup cheese and 2 tablespoons chopped poblanos into center of dough. Close palm to seal edges of dough and place on a clean work surface with a long side of husk facing you. Roll husk away from you to form a tight log. Fold stem end of husk over and tie with a strip of corn husk. Twist thin end of tamale and tie with another strip of corn husk. Repeat this process to form remaining tamales.
Insert a steamer basket inside a large pot with 1/2 inch boiling water. Place tamales in basket, cover pot, and steam tamales for 1 hour, checking pot occasionally and adding hot water as needed to prevent scorching the bottom of the pot.
Serve tamales with salsa verde.