Preheat the oven to 375° and butter a 9-by-13-inch glass or ceramic baking dish. Arrange one-third of the bread in the bottom of the dish and spread with half of the mustard. Top with half of the ham, 1 tablespoon of the tarragon and one-third of the Gruyère cheese. Repeat the layering, 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. Pat the remaining cheese on top and cover with a sheet of buttered parchment paper.
Bake the strata in the center of the oven for 30 to 35 minutes, until bubbling and browned around the edges. Remove the parchment and turn on the broiler. Broil for about 3 minutes, just until the top is golden and bubbling. Let rest for 10 minutes, then cut the strata into squares and serve.