Preheat the oven to 450 degrees . In a large skillet, melt the butter over medium heat. Add the mushrooms and thyme and season with salt. Cook, stirring frequently, until the mushrooms are browned and their juices cooked off, about 10 minutes. Set aside to cool.
Place a cooling rack on top of a baking sheet and set aside. Place the eggs and flour in separate shallow bowls; lightly beat the eggs. In another shallow bowl, combine the mushroom mixture with the panko.
On a work surface, pound the chicken 1/2 inch thick and cut into 1-by-4-inch pieces. Sprinkle each piece with salt, then coat with flour and pat off the excess; dip into the egg and roll in the mushroom mixture. Place the chicken fingers on the prepared baking sheet and bake until golden and cooked through, 10 to 12 minutes.