Butter a 2-quart shallow baking dish (oblong or rectangle) and set aside. In a bowl, combine shredded cheeses and reserve. In baking dish, stand a few bread slices upright with cut side down. Add ham and a sprinkling of cheese between each slice, continuing until all has been used and fitted.
In a large bowl, whisk together eggs, half and half, mustards, salt and pepper. Gently pour custard over the bread, tucking ham between the layers to keep moist during baking. Cover tightly with plastic wrap and chill at least 4 hours up to overnight. (Overnight is best; the longer the bread rests and absorbs the custard, the lighter and creamier the texture will be after baking.)
Preheat oven to 350 degrees. Place baking dish onto a baking sheet (to catch drips) and remove plastic wrap. Rest at room temperature, approx. 1 hour. Sprinkle with chives and bake 1 hour until puffed and set in center (gently press on center; if milky liquid appears, continue baking). If strata browns too quickly, tent loosely with foil and continue cooking until set. Let stand 10 min before cutting and serving.
* You'll need about 8 full slices. Your Market Bakery can slice artisan loaves for you, upon request.