Farmhouse Cheese and Caraway Soda Bread Puddings

Preheat oven to 375°F with rack in middle.
Whisk together flour, baking soda, salt, and caraway in a large bowl. Add buttermilk and butter. Stir just until dough is evenly moistened but still lumpy.
Transfer dough to a well-floured surface and turn dough to coat with flour. Gently knead with floured hands about 8 times to form a soft but slightly less sticky dough.
Form into a domed 6-inch round and put on an ungreased baking sheet. Cut a 1/2-inch-deep "X" across top of each loaf with a sharp knife. Using a small sieve, dust 1 tablespoon of additional flour over loaf.
Bake in middle of oven until golden brown and bottom sounds hollow when tapped, 35 to 40 minutes. Transfer loaf to a rack and cool completely.
Preheat oven to 350°F with rack in middle. Butter ramekins or baking dish.
Cut enough bread into 1-inch cubes to measure 3 3/4 cups (about half the loaf; reserve remaining bread for another use.) Divide bread among ramekins (or spread in larger baking dish).
Whisk together eggs, milk, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Stir in cheese and spoon over bread in ramekins or baking dish. Let stand 10 minutes.
Put ramekins or baking dish in a larger baking pan and add enough hot water to come halfway up sides of ramekins or dish.
Bake until just set and a knife inserted in center comes out clean, about 35 minutes.
Transfer ramekins or dish from hot water bath to a rack and cool 20 minutes before serving.