Add the sugar and water to a heavy-bottomed saucepan. Swirl to start the sugar dissolving. Place over a high flame or high heat and bring to a boil. Do NOT stir this. Swirl it when necessary, but don't stir! Cook like this, swirling every so often, until it reaches a deep orange brown colour. Remove the pan from the heat and add the heavy cream all at once. Be careful, as it will splatter and spit quite a bit.

At this point you're going to think you've done something terribly wrong. The caramel you coaxed along so carefully will seize up in a horrid lump in the middle of the cream. Don't worry. It will all come out right in the end!

Place the pan back on the burner and reduce the heat to medium. NOW you can STIR! Stir frequently to help melt the caramel into the cream. It will bubble, that is okay! After about 5 minutes, the caramel should be fully melted into the cream. Remove the pan from the heat and stir in the rum and cold butter pieces until the butter is fully melted and incorporated.

Remove the pan from the heat and cover it, letting it cool, while you prepare the Banana Churros.

Peel and mash the banana until totally smooth. Scrape this into a liquid measuring 2-cup measuring cup and fill with water to the 1 1/4 cup line. Stir together and pour into a heavy-bottomed 3 quart saucepan. Add the butter and salt and bring to a rolling boil. Add the flour all at once, stirring vigorously to combine, over LOW heat. Continue stirring until the mixture forms a ball, about 1 minute. You'll know you've done it right if there's a little film on the bottom and sides of your pan.

Remove the pan from the heat and transfer the dough to the bowl of a stand mixer fitted with a paddle or a heat-proof bowl you can use with a hand mixer. Let it rest for 5 minutes.

While the dough rests, bring 1-inch of peanut or canola oil, or lard, up to 375°F in a heavy-bottomed, high-sided frying pan.

Back at the mixer, add the eggs in all at once and beat on high speed until fully incorporated and the dough holds the shape of the beater or paddle in it.

Scrape the dough into a large pastry bag fitted with a large star (1/2-inch opening) or a large blank tip (1/2-inch opening). Twist the top of the pastry bag and use a twist-tie to cinch it shut. Carefully pipe 4-inch strips into the hot oil. Use a blunt side of a butterknife to help release the strips from the pastry tip if necessary. I almost always need to do this. Only pipe 4 or so strips at a time, do not crowd the pan because they will expand as they fry!

Fry them until golden brown for about 2 minutes per side, turning with tongs when necessary. Drain the churros on paper towel lined plates. Sprinkle lightly with cinnamon sugar and serve with a bowl or squeeze bottle of Butter Rum Caramel.