Grease the bottom and sides of the pan thoroughly with the shortening or butter, then dust with enough powdered sugar to coat the pan in a thin layer. This will keep the marshmallow from sticking to the pan.
Pour 1/3 cup of the water into the bowl of your stand mixer. Sprinkle the gelatin over the water, but don't stir it in. Leave it to sit while you complete the next step.
In a saucepan, stir together 1/3 cup of water with the sugar, corn syrup and salt. If there are any grains of sugar on the sides of the pan, brush them off with a wet pastry brush.Over medium heat, cook the syrup (without stirring) until it reaches 240 degrees Fahrenheit. You can check the temperature with a candy thermometer or a digital instant read thermometer. This step might take around 10 minutes. Turn the mixer on to low speed, and slowly drizzle the boiling syrup into the gelatin mixture. Increase speed to medium (#6 on a Kitchen Aid stand mixer), and whip for about 10-11 minutes.The marshmallow should now be pure white, lukewarm, very sticky and thick, but not so thick that you won't be able to spread it into a pan. Beat in the vanilla extract and vanilla bean paste.
Using a greased spatula, immediately scrape the marshmallow into the prepared pan. Lightly grease your hands with butter or shortening and press the marshmallow out evenly in the pan. Lightly sprinkle the top of the marshmallow with more powdered sugar.
Cover the pan with a clean kitchen towel, and let sit at room temperature for 6 hours (or overnight) before cutting.
To cut the marshmallows, turn the pan over onto a cutting board. The whole slab should fall out of the pan onto the cutting board if you prepped the pan with shortening/butter and powdered sugar.
Grease a sharp chef's knife with butter or shortening. Cut the marshmallow into 5 strips, then cut each strip into 5ths, to get 25 large marshmallows. If you like, you can cut each of those into 4ths, to get 100 small marshmallows. You'll need to grease the knife again in between cuts. As you cut the marshmallows, dust the cut sides with powdered sugar to keep them from sticking.
Store the marshmallows in an airtight container at room temperature. They will stay soft and fresh for about 2 weeks.