8 poblano chile peppers (4 1/2 to 5 inches long)
12 ounces muenster cheese, cut into 8 sticks (about 3 1/2 by 1 inch)
1 1/2 pounds tomatoes, cored and chopped
1 small onion, chopped
2 cloves garlic, chopped
3 tablespoons extra-virgin olive oil
2 whole cloves
7 black peppercorns
2 small bay leaves
1 cinnamon stick
1 sprig fresh cilantro, plus leaves for garnish
4 cups low-sodium chicken broth
Vegetable oil, for frying
6 large eggs, separated
All-purpose flour, for dredging
Preheat the broiler. Place the chiles on a foil-lined broiler pan; broil, turning, until the skins char, about 8 minutes. Place in a bowl, cover with plastic wrap and let cool, about 10 minutes.
Peel the chiles. Slice each open lengthwise, about 1/2 inch from the top to 1 inch from the bottom. Remove the seedpod, then rinse to flush out any remaining seeds. Blot dry, inside and out, with paper towels. Carefully stuff each chile with a stick of cheese.
Puree the tomatoes, onion and garlic in a blender until smooth. Heat the olive oil in a large skillet over high heat until almost smoking. Add the tomato puree, cloves, peppercorns, bay leaves, cinnamon stick and cilantro sprig and fry, stirring, until thick, 5 to 6 minutes. Add the broth, reduce the heat and simmer until thickened but still brothy, 25 to 30 minutes. Season with salt. Strain, discarding the solids, and keep warm.
Preheat the oven to 350. Line a baking sheet with paper towels. Heat 1 to 1 1/2 inches of vegetable oil in a large pot over medium-high heat until a deep-fry thermometer registers 360. Beat the egg whites, 2 tablespoons water and 1 teaspoon salt with a mixer until stiff but not dry. Beat the yolks in a separate bowl until combined, then gently fold them into the whites.
Overlap the sides of the slit in the chiles to enclose the cheese, then dredge in flour, working with two chiles at a time. (Coat any tears with flour.) Using tongs, dip and roll the chiles in the egg mixture to coat. Fry in the hot oil until golden, 2 to 3 minutes per side. Transfer with a large skimmer or 2 spatulas to the paper-towel-lined baking sheet, letting the excess oil drip back into the pot. Repeat with the remaining chiles and egg mixture. Place in the oven on the lined baking sheet to cook off any excess oil, about 5 minutes.
Ladle the tomato broth into 4 shallow bowls; place 2 chiles in each. Garnish with cilantro.