In a heavy saucepan, combine molasses, granulated sugar, brown sugar and water. Heat to boiling and stir to dissolve sugars. Let mixture boil, without stirring, until syrup reaches the hard ball stage, 260°F on a candy thermometer. Remove from heat.
Stir in butter, baking soda, and salt. Immediately pour onto a buttered marble slab or baking sheet. When cool enough to handle, butter hands and gather taffy into a ball. Pull taffy between hands and continually stretch and fold again until taffy turned lighter in colour. This can take up to 15 minutes.
Stretch and twist taffy into a rope about 1 inch thick and cut into pieces using buttered scissors. Wrap each piece in waxed paper.