Preheat oven to 425 degrees. In a medium saucepan over medium-high heat, melt butter until foaming. Whisk in flour and cook for 1 minute, stirring constantly.
Stir in enchilada sauce, chicken broth, and Sriracha. Bring to a boil; reduce heat and simmer until reduced by half, about 20 to 25 minutes. Season to taste with salt and pepper. Keep warm over low heat.
Meanwhile, in a medium saucepan or skillet, heat oil until shimmering. Add onions and green chiles and heat until the onions have softened without browning, about 5 to 7 minutes.
Add the chicken, sour cream, and 4 ounces of Sriracha Gouda. Stir until the filling is heated through and the cheese is creamy, about 2 minutes.
Spread 1/2 cup sauce in the bottom of an 8-1/2 by 11-inch rimmed baking dish.
Working with 1 tortilla at a time, top with a heaping 1/4 cup of the chicken filling. Using a spoon, shape the filling in to a log. Roll the tortilla around the filling and lay it seam-side down in baking dish.
Fill remaining tortillas. Do not cover with sauce yet. Bake until the tortillas have begun to crisp and turn light brown, about 10 minutes. Remove from oven.
Pour the remaining sauce over the enchiladas and sprinkle with the remaining 4 ounces Sriracaha Gouda. Bake until the cheese is hot and bubbly, about 10 minutes longer
Let stand 5 minutes. Garnish with scallions and serve with additional Sriracha if desired.