Place both sugars, corn syrup, and water (and rum if using) in a saucepan. Stir gently until the sugar is moistened and mixed with the water.
Heat over medium-high heat while swirling the pot occasionally to dissolve the sugar. Allow the sugar syrup to come to a boil, with all the sugar dissolved.
Cover the saucepan with a lid and lower the heat to medium. Let the syrup cook for 2 minutes. Do not lift or remove the lid at this point. (The condensation that forms on the bottom of the lid will help wash away any undissolved sugar crystals on the sides of the saucepan back into the syrup).
After 2 minutes, remove the lid and check to make sure there are no undissolved sugar crystals on the sides of the saucepan. If there is any, then cover the saucepan for a further minute to allow the condensation to wash down the sugar crystals into the syrup.
Clip the sugar thermometer to the side of the saucepan, and set the temperature to between 242 - 245°F. Let the sugar syrup cook until it reaches this temperature - do not exceed 245°F. Make sure to swirl the saucepan to evenly distribute the heat. Then swirl the saucepan one last time, after the right temperature is reached as well (to make sure the sugar syrup is evenly hot throughout).
If you don't have a sugar thermometer, please see this homemade marshmallows post for directions to check the correct stage of the sugar syrup.
When the sugar syrup has reached the correct stage, remove from the heat and let it cool down a little, and the bubbles subside. Then, with the mixer running on medium - low speed, pour the sugar syrup into the gelatin, along the wall of the bowl (this is to prevent the gelatin from being scorched by the hot sugar syrup). Then increase the speed to medium - high.
Whisk on medium - high for 3 to 5 minutes until the marshmallow base doubles in size and is starting to become fluffy and smooth. Mix for a further 3 to 5 minutes on high speed, allowing the marshmallow to volumize more (up to 3 times), and cool down to the touch. Stop mixing if the marshmallow base has already cooled down, as this will create hard to handle marshmallows. Make sure the base is fluffy, pale brown and up to 3 times the initial volume of the sugar syrup.
Add the vanilla and rum in the final 2 minutes of whisking, so that they get incorporated well.
While the marshmallow base is whisking, prepare the pan and utensils that you'll be using. Rub an 8 x 8 inch pan or a 9 x 9 inch pan with butter (or you can line this pan with parchment paper if you prefer). Brush all the utensils - spoon, spatulas with butter as well.
Once the marshmallow base has tripled in size, remove the bowl from the mixer and take about 1 cup of the marshmallow mix and add it to the bowl with the melted lukewarm butter. Mix until the butter is well incorporated in the marshmallow mix.
Evenly pour this butter and marshmallow mix back into the rest of the marshmallow base. Gently fold it into the marshmallow base to create ripples. OR you can fold it in further to completely mix it in with the marshmallow bases so that you don't end up with ripples/swirls.
Scrape the marshmallow base into the prepared pan using a silicone spatula or flexible scraper. Spread the marshmallow evenly in the pan.
Generously sieve mallow dusting powder over the marshmallow and let it set for at least 6 hours.
Turn the set marshmallow out onto a work surface, dusted with dusting powder. Cut the marshmallows with an oiled knife or a pair of scissors into 25 pieces, or 16 pieces for larger marshmallows. Toss the marshmallow in the dusting powder and store in an air-tight container.