Preheat oven to 400°F. Line a baking sheet with parchment paper, followed by layer of mini marshmallows.
Heat half and half cream for 90 seconds (or until warm) in microwave and set aside.
Add sugar to saucepan, shaking to evenly cover bottom. Place over medium heat, stirring gently once sugar begins to dissolve (watch closely to avoid burning). Continue to heat and stir until sugar is completely dissolved and syrupy.
Immediately remove from heat and slowly pour in about a third of the warmed half and half. Place back onto medium-low heat and stir until sugar dissolves, about 3 minutes (there may be small bits, but that's okay). Stir in remainder of half and half. Turn to low, stirring occasionally.
Place marshmallows in oven and bake until puffy and lightly golden, about 5 minutes. Be sure to watch closely as they will burn quickly.
Remove from oven and immediately transfer to saucepan (with cream mixture) using a spatula. Turn heat back up to medium-low and continue to stir, using back of spoon to break up marshmallows. Remove from heat once marshmallows dissolve and mixture is creamy and frothy.
Pour over fine strainer to remove any bits, then add vanilla. Cool, then transfer to jars. Store in refrigerator and use within 2 weeks.