In a small skillet or saucepan over medium heat, combine the blackberries, 2 Tablespoons granulated sugar and lemon juice. Stir until sugar is dissolved. When the mixture boils, reduce heat to low and simmer for 10 minutes, stirring occasionally and mashing the blackberries.
Transfer the compote to a bowl and chill in the refrigerator for at least 30 minutes*. The mixture should be fully cooled before using for the next step.
Make the marshmallows
Grease a 9x9-inch baking pan with cooking spray and set aside.
In the bowl of a stand mixer, combine blackberry compote and ½ cup cold water. Sprinkle with gelatin and let bloom.
In a medium saucepan over medium-high heat, combine granulated sugar, corn syrup, salt and ½ cup water. Stir until sugar is dissolved. Bring the mixture to a boil, swirling the pan occasionally, until the mixture reaches exactly 240°F on a candy thermometer (about 10-12 minutes). Immediately remove from heat.
With the stand mixer on low speed, slowly and carefully pour the melted sugar mixture into the gelatin mixture. Add the vanilla extract.
Slowly (to avoid splashes) increase the speed to high and whip until the mixture is stiff and pale purple, about 8-10 minutes. When you remove the whisk, stiff peaks should form that take a few seconds to fall back into the marshmallow mixture. Working quickly, pour the mixture into the prepared baking pan and use a spatula to help spread.
Let the marshmallows firm for at least 4 hours at room temperature or 2 hours in the refrigerator.
Lightly sprinkle your work surface with powdered sugar. Run a butter knife around the edges of the pan, then turn the marshmallow out onto the work surface. You may need to pry it out at a corner. Use a sharp knife to cut the marshmallow into 16 pieces. If needed, spray the knife with cooking spray to help prevent sticking. If there is excess cooking spray on your marshmallow from the knife, dab with a paper towel to remove.
If using, roll each marshmallow in powdered sugar on each side to prevent sticking, then serve.
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Notes
*If you are concerned about the seeds in the blackberries, you can purée the compote after it has cooled. If needed, strain the berry mixture to remove any seeds. Corn syrup substitute: You can use honey or agave nectar in a 1:1 substitute for the corn syrup. Gelatin substitute: There is not an easy substitute for gelatin. Agar agar cannot be used as a 1:1 substitute.