Bread Pudding: Preheat oven to 350F. Whisk eggs in a bowl and combine with 4 cups of the milk. Add remaining ingredients. NOTE: The brown sugar is optional and really only useful if you're not making the Rum Sauce - otherwise it is too sweet!) Gently and slowly pour half the milk into the mixture, stirring until all bread is moistened. Add remaining milk as needed to reach a soft batter texture. (You will use most if not all of the quart.) Lightly grease a 13x9 pan and pour mixture into pan. Bake for 60-75 minutes (depending on pan). Dish is done when knife inserted in center comes out clean. Rum Sauce: Soften butter, add sugar and cream together well on medium speed, be sure the sugar is well creamed and nearly smooth or the sauce will come out "grainy" tasting. Add eggs at low speed, gradually increasing speed to high. Add vanilla and beat on high for 20-25 minutes, stirring and scraping sides of bowl every few minutes. Do not skip this step or your texture will come out wrong. Mixture will become light and foamy, add rum and mix on medium until well blended. Pour into container to be refrigerated until served.