Lightly grease a 9x13 baking dish and spread half the bread cubes in an even layer across the bottom.
Scatter half the diced ham and half the shredded cheddar over the bread layer.
Add the remaining bread cubes on top, followed by the rest of the ham and cheddar.
In a large mixing bowl, whisk together the eggs, whole milk, heavy cream, Dijon mustard, garlic powder, onion powder, salt, and black pepper until fully combined.
Pour the custard mixture evenly over the layered bread and filling, pressing the bread gently to help it absorb the liquid.
Cover the dish tightly with plastic wrap and refrigerate for at least 6 hours, or overnight.
When ready to bake, remove the dish from the refrigerator and let it sit at room temperature for 20 to 30 minutes while the oven preheats to 350°F (175°C).
Bake uncovered for 45 to 50 minutes, until the top is deep golden brown, the edges are set, and the center no longer jiggles when gently shaken.
Let the strata rest for 10 minutes before slicing and serving, topped with fresh chives if desired.