Into a stand mixer, mix together the milk, instant dry yeast, brown sugar, eggs, butter and salt. Switch to the dough hook, add flour and mix until a dough forms. Cover the bowl with a towel and rest for 1 hour.
In a saucepan, combine the brown sugar and butter. Bring to a boil over medium heat and simmer for 4 minutes. Remove from heat, stir in rum. Pour mixture into the baking dish and top with the chopped pecans. Set aside.
Mix together brown sugar and cinnamon until all is combined. Set aside. Transfer the dough onto your floured surface and roll into a rectangle. Brush on melted butter. Sprinkle the sugar mixture over the melted butter. Roll the dough into an 18-inch long log.
Using a sharp knife, cut into 12 rolls. Transfer the rolls on top of the rum toffee pecan topping in the baking dish. Cover the rolls with a towel and let them rise for 30 minutes.
Bake at 375° for 25-30 minutes or until golden brown on top. At the 12 minute mark, cover the top of the rolls with aluminum foil to avoid them browning too much. Remove from oven and let them cool for 5 minutes.