Place sliced potatoes in a large bowl and cover with cold water. This removes excess starch and prevents browning while you prep everything else.

Heat 2 tablespoons butter in a large skillet over medium-high heat. Season chicken with 1/2 teaspoon salt and the pepper. Add chicken and cook until golden on the outside, about 5 minutes. It does not need to be fully cooked through. Transfer to a plate. Add mushrooms and onion to the same skillet. Cook until mushrooms release their liquid and start to brown, about 8 minutes. Add garlic and thyme; cook 1 minute more. Transfer to the plate with chicken.

In a medium saucepan, melt remaining 2 tablespoons butter over medium heat. Whisk in flour and cook, stirring constantly, for 2 minutes until golden and bubbling. Slowly whisk in chicken broth and milk. Simmer, whisking often, until thickened enough to coat the back of a spoon, about 5 minutes. Remove from heat and stir in half the Gruyere until melted. Season with remaining 1 teaspoon salt.

Preheat oven to 375F. Drain potatoes and pat dry. Spread half the potatoes in an even layer in a greased 9x13 inch baking dish. Top with half the chicken-mushroom mixture. Pour over half the sauce. Repeat layers once more, ending with sauce. Cover tightly with foil.

Bake covered for 40 minutes. Remove foil, sprinkle with remaining Gruyere, and bake uncovered until potatoes are tender when pierced with a knife and the top is golden and bubbling, about 15 minutes more. Let rest 10 minutes before serving so the sauce sets.