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Place diced potatoes in a small pot, cover with water, bring to a boil, and cook 5-7 minutes until fork-tender. Drain and set aside.
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Preheat oven to 375°F. In a large pot, melt butter over medium heat. Add onion and celery, cook 4-5 minutes until soft. Add garlic and cook 30 seconds.
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Sprinkle flour over the vegetables and stir constantly for 1 minute to make a roux.
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Slowly pour in chicken broth while stirring, then add milk. Keep stirring until the mixture thickens to coat the back of a spoon, 3-4 minutes.
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Stir in shredded chicken (cooked to 165°F internally and verified with meat thermometer), parboiled potatoes, frozen peas and carrots, salt, pepper, and thyme. If using rotisserie chicken (which is already cooked to 165°F), simply shred and use. Verify it's still warm; if cold, warm it gently in the sauce before proceeding.
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Pour hot filling into a greased 9x13 baking dish. Unroll crescent dough, separate into triangles, and arrange across the top overlapping slightly. Brush with beaten egg and sprinkle with flaky salt.
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Place on a sheet pan to catch overflow. Bake 20-25 minutes until crust is golden brown and filling is bubbling at the edges.
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Let rest 5-10 minutes before serving — the filling is volcanic when it first comes out. Spoon out generous portions.