Spread the marshmallows out onto heat-proof plates.
Using a blow torch, roast the mini marshmallows to desired burntness. Set aside.
Combine milk, heavy cream, vanilla, salt and 1/4 cup sugar in a large saucepan.
Heat the cream and sugar mixture over medium-low heat, stirring until sugar has dissolved.
Add eggs, egg yolk, and 1/4 cup sugar to a mixing bowl and beat until thick.
Temper the eggs with the heated milk/cream mixture but adding a little bit of milk to the eggs while whisking.
Once you have the milk/cream and eggs incorporated, add it all back to the saucepan and heat over medium heat until thickened slightly.
Add the marshmallows to the custard and stir until slightly melted. Use a stick blender to smooth out the base.
Pour the custard into a bowl and cover with plastic wrap.
Allow the custard to chill overnight.
Churn the ice cream base in your ice cream maker according to manufacturers instructions.
Transfer to a container and store in the freezer.