Preheat the oven to 400°F and position a rack in the lower-middle slot.
In a large deep ovenproof skillet or Dutch oven, melt 4 tablespoons unsalted butter over medium-high heat.
Whisk in 1/2 cup all-purpose flour to form a smooth paste and cook briefly, about 1 minute, stirring to remove the raw flour taste.
In a medium bowl, whisk together the 14.5-ounce can chicken broth, 1 1/2 cups salsa verde, 1/2 cup green taco sauce, and the 5-ounce can evaporated milk until combined.
Gradually whisk the liquid mixture into the skillet, bring to a simmer, and cook, whisking occasionally, until the sauce thickens.
Stir the shredded rotisserie chicken (about 6 cups) into the sauce and heat through; cover the pan to keep warm while you prepare the topping.
Combine 3 tablespoons unsalted butter, 1 cup whole milk, 1 1/2 cups all-purpose flour, 1/2 cup cornmeal, 1 tablespoon baking powder, 3/4 teaspoon salt, and 1/4 cup sliced green onions in a bowl to form a biscuit-style batter; stir in the 1/4 cup chopped cilantro.
Spoon or drop the batter evenly over the chicken mixture in the skillet to create a crust, leaving some gaps so steam can escape.
Bake in the preheated oven until the topping is golden and cooked through, about 20–25 minutes.
Remove from the oven and let rest a few minutes before serving.
