Preheat your oven to 350°F (175°C).
Place the bread cubes in a large mixing bowl. Pour the milk over the bread and let it soak for about 15 minutes, stirring occasionally.
In a small skillet, melt the butter over medium heat. Add the finely chopped onion and sauté until it becomes soft and translucent.
In the bowl with soaked bread, add the cooked onions, grated Gruyère cheese, grated Emmental cheese, dried thyme, black pepper, and salt. Mix well to combine.
Stir in the beaten eggs until everything is evenly mixed.
Transfer the mixture to a greased baking dish or casserole dish.
Bake in the preheated oven for about 30-35 minutes, or until the top is golden brown and the dish is set.
Allow the Kachkéis to cool slightly before serving. It can be enjoyed warm or at room temperature.
Enjoy your meal!