Heat the butter over medium-high heat in a large, deep ovenproof skillet (or a 5 to 6 quart Dutch oven). Whisk in the flour to make a paste.
In a medium bowl, whisk together the broth, salsa, taco sauce and evaporated milk; whisk into the pan. Bring to a simmer, whisking occasionally, until the sauce thickens. Stir in the chicken, heat through, and cover to keep warm.
Meanwhile, adjust the oven rack to the lower-middle position and heat the oven to 400°F.
PREPARE THE DUMPLINGS:
In a small saucepan, heat the milk and butter until just about to simmer.
In a medium bowl, stir together the flour, cornmeal, baking powder, salt, green onions and cilantro. Stir in the warmed milk mixture, just until blended.
Scoop rounded spoonfuls of the dough and drop into the chicken mixture. Return the chicken to a simmer over medium-high heat.
Cover and transfer the pan to the oven and bake until the dumplings are cooked through, 15 to 20 minutes.
Notes
It's completely up to you how spicy you'd like to make this recipe. Choosing mild or medium-spiced salsa will determine that.