Bring a large pot of lightly salted water to a boil. Cook egg noodles in boiling water, stirring occasionally until almost cooked through and still firm to the bite, about 4 minutes. Drain.
Melt butter in a large saucepan over medium heat. Whisk in flour; cook, stirring, until mixture is light golden, about 3 minutes.
Pour milk into butter and flour mixture, whisking constantly. Stir in cream of mushroom soup, green onions, seasonings, and dried tarragon. Bring mixture to a simmer, cook for 2 minutes. Remove from heat.
Stir in pepperjack cheese until cheese is melted and incorporated.
Stir pepperjack cheese mixture, turkey or chicken meat, and noodles in a large bowl until evenly incorporated. Pour mixture into a large casserole dish. Sprinkle onion rings on top, pressing down slightly with the tines of a fork. Bake in the preheated oven until casserole top is golden brown and sauce is bubbling, about 30 minutes.