Preheat oven to 375° and butter a 9 X 13 glass or ceramic baking dish.
Arrange 1/3 of the bread in the bottom of the dish and spread with half the mustard. Sprinkle half of the ham and optional veggies, 1 T of the tarragon and 1/3 of the Gruyère cheese. Repeat the layers ending with the third layer of bread and leaving off the last layer of cheese.
In a medium bowl, whisk the milk with the eggs and season generously with black pepper. Pour the custard evenly over the dish, pressing the bread to absorb the liquid.
Cover with waxed paper and foil and refrigerate 8 hours or overnight.
Pat the remaining cheese on top and cover with a well-buttered (!!) piece of parchment paper. Let come to room temperature (@ 45 min) before baking. Bake until bubbling and browned around the edges. Remove the parchment and turn on the broiler. Broil for @ 3 minutes, just until the top is golden and bubbling.
Let rest 10 minutes before serving.
Makes 10-12 servings.
For more information, see New Brunch Favorite: Monte Cristo Strata.