For the cinnamon buns: In a pot, combine the milk and the butter and bring to a simmer to melt the butter. Remove from the heat and cool the mixture to 120 degrees F. Grease a bowl with oil.
In a stand mixer fitted with a dough hook, combine the cooled milk mixture, 1 cup of the flour, the sugar, salt, egg and yeast. Mix with a dough hook on low for 3 minutes. Add the remaining 2 1/2 cups flour and mix on low until the dough forms a ball and pulls away from the sides of the bowl. Turn the speed of the mixer up to medium-high and knead the dough for 8 minutes.
Transfer the dough to the greased bowl and cover with plastic wrap and kitchen towel. Place the dough in a warm place and allow it to proof for 2 hours.
For the filling: Preheat the oven to 375 degrees F. Turn the dough out onto a floured work surface and roll out to a 1/4-inch thick sheet. Spread the butter over the surface of the dough leaving about a 1/2-inch border around the edges uncovered.
In a bowl combine the brown sugar, granulated sugar and cinnamon. Spread an even layer of the filling on top of the buttered dough. Roll up the sheet of dough into a log. With the seam-side down cut the log into 12 to 16 even portions. Place the portions in large muffin tins and allow them to proof again until the buns fill the muffin tin.
Bake for 20 to 30 minutes.
For the pecans: In a pan, combine the brown sugar and 2 tablespoons water, and cook until the water is cooked off and the sugar is melted.
In a bowl, toss the pecan pieces with the melted sugar. Spread the coated pecans on a sheet pan and cool. Break up the candied pecans once they are cooled, and reserve for later use.
For the caramel sauce: In a pot over medium heat, combine the sugar and corn syrup. Stir regularly until the sugar begins to melt, about 10 minutes. Once melted, stir the sugar until it reaches a golden-caramel color. Remove the caramel from the heat and slowly stir in the heavy cream. Stir in the butter and hold.
In a saute pan, heat the oil. Add the apples and saute until tender. Add the rum and flambe. Add the apples to the caramel sauce and simmer to marry the flavors, 2 to 3 minutes.
To serve, warm 6 of the cinnamon buns and place the one bun in the center of each plate. Spoon the apple-rum caramel sauce over the cinnamon buns. Sprinkle the candied pecans over the caramel sauce. Top the dressed cinnamon buns with a scoop of vanilla ice cream.
This recipe makes 12 to 16 cinnamon buns, reserve any extra for another use.
Use extreme caution when igniting alcohol. Remove the pan from the heat source before adding the alcohol. Pour the alcohol into the pan and carefully ignite with a match or click lighter. Return the pan to the heat and gently swirl to reduce the flames.