Grease a 9x13-inch baking dish thoroughly.
Place half of the cubed bagels into the dish. Scatter half of the cream cheese cubes and half of the sliced red onion evenly over the bread.Add the remaining bagel cubes on top, followed by the rest of the cream cheese. In a large bowl, whisk together the eggs, whole milk, heavy cream, Dijon mustard, pepper, and just a small pinch of salt.Pour the custard evenly over the layered bagels. Use clean hands or a spatula to gently press the bread down so it absorbs the liquid. Cover the dish tightly with foil or plastic wrap and refrigerate for at least 2 hours. Overnight chilling is ideal for the most tender, custard-like texture.
Preheat the oven to 350°F (175°C).Bake the strata covered with foil for 30 minutes. Remove the dish from the oven and carefully take off the foil.Gently tuck the smoked salmon pieces into the top layer of bagels.Sprinkle the shredded Havarti or Swiss evenly over the top. Return the dish to the oven uncovered and bake for another 15–20 minutes.
Allow the strata to rest for 10 minutes before serving.Top generously with fresh dill, chopped chives, and any remaining red onion slices.
