Combine the sugar, water, and corn syrup in a small, heavy saucepan and place over high heat. As soon as the sugar has dissolved, heat your thermometer under warm tap water (to prevent thermal shock) and attach to side of pan, making sure that the bulb is completely submerged. Do not stir the mixture or agitate the pan at this stage.
When the mixture reaches 230 degrees, reduce the heat to medium and allow the sugar to color gradually. As the temperature approaches 300 degrees the color will start to turn to amber. At this point, it's safe to swirl the pan gently to help distribute the heat evenly. As the caramel heads towards 340 degrees, it will reach a deep amber color.
Remove the pan from the heat and slowly stir with a metal spoon, allowing it to cool until the caramel falls from the spoon in a solid stream, like a string. When it does, it's doodad time.
Form doodads by holding the spoon 12 to 16 inches over the caramel and drizzling the patterns on the parchment. (Hint: allow the spoon to unload into the pan for a few seconds before attempting doodad. This will allow a narrower stream.) Quick back and forth motions or side to side movements are best. It's all in the wrist. Allow doodads to cool for 15 minutes then peel the paper off (not the other way around). Stack in an airtight container with wax paper between each piece.
After making 15 to 20 doodads, you should have sufficient caramel remaining to move on to the sauce phase. So, return the remaining caramel to medium high heat and continue cooking until you see the first sign of smoke. Immediately remove from the heat, and add the cream, all at once, at arms length. (It will boil furiously.) Once the mixture calms down, return to medium heat and boil 3 more minutes, stirring occasionally. Cool to room temperature, move to squirt bottle and refrigerate for up to a week.
To make the bananas brulee, quarter the bananas (peel on) and rub into sugar to coat thoroughly. Remove peel and place on cooling rack over several layers of foil. (Don't try this on your dining room table.) Don your safety goggles and fire up your torch. Hold the torch so that the very tip of the flame barely touches the banana, and move quickly back and forth until the sugar melts, turns brown, and bubbles. As soon as it looks like caramel, move on. You know you've got it down when a solid, glasslike sheet of gold (no graininess) has formed on the banana.
To do the split (so to speak), squirt a pattern of caramel sauce on a chilled plate, top with 4 banana pieces (Lincoln log style), top with the ice cream of your choice (we like our peachy vanilla from Churn Baby Churn), and finally, a doodad.