Preheat oven to 350 degrees F. Line a baking sheet with parchment paper or a silicone baking mat and put it on the center oven rack.
Whisk together the whole wheat flour and wheat/oat bran or wheat germ in a large bowl.
In a medium bowl, whisk together the all-purpose flour, baking soda and salt. Add to the wheat flour mixture and whisk to combine.
Add the butter pieces and rub them into small pieces with the flour mixture using your fingers or a pastry blender, until as small as possible.
Stir in the buttermilk and molasses until the dough is uniformly damp. Turn out onto a lightly floured countertop and knead gently, until the dough forms a smooth ball.
Use a sharp serrated knife (or a lame) to slice a cross deeply into the top of the bread, about 1-inch deep. Place the loaf on the hot baking sheet.
Bake for about 30 to 35 minutes, or until the loaf is firm on top and when you tap the bottom, feels hollow.
Remove from the oven and cool on a wire rack for about one hour before serving.
Buttermilk: If you do not have buttermilk and can't get it, use this substitution: Place 1½ tablespoons lemon juice or white vinegar in a liquid 2-cup measuring cup. Add enough whole or 2% milk to bring the mixture to 1½ cups. Stir it, then let it sit for 5 minutes before using. Discard ¼ cup to get the correct measurement for this recipe.
Storing: Store fresh baked Irish brown bread in an airtight container at room temperature for up to 3 days after baking.
Freezing: If you don't plan on eating your bread right away I suggest freezing the loaf. Wrap the cooled loaf tightly in two layers of plastic wrap and freeze for up to 3 months.
Reheating: When you are ready to reheat the loaf, remove it from the freezer and allow it to come to room temperature. Pop in the oven and bake at 350°F for 5-10 minutes.