Preheat the oven to 375°F (190°C). Grease a 9x13-inch casserole dish.
In a large Dutch oven, melt butter over medium heat. Add onion and cook until tender, about 5 minutes, stirring often.
Stir in flour, thyme, mustard, garlic powder, onion powder, salt, and pepper. Cook, stirring constantly, for 1 minute.
Gradually mix in milk and broth. Cook, stirring constantly, until the mixture bubbles and thickens, 3 to 5 minutes.
Stir in chicken, vegetables, and parsley. Remove from heat and cover to keep warm.
On a cutting board, unroll crescent dough sheets. Cut each sheet into 12 (1-inch) strips, creating 24 strips total. Roll each strip into a spiral.
Pour the chicken mixture into the prepared casserole dish. Top with dough spirals.
Bake until the filling bubbles around the edges and the crescent dough turns golden brown, 20 to 25 minutes.
Let stand uncovered for 15 minutes before serving.
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