Set a rack in the oven on the second position from top and turn the broiler to low (or on if only one setting is available). Line a cookie sheet with a silicone mat and spread the marshmallows out in one layer on their sides.
Place the cookie sheet under the broiler, keep the door ajar and a solid eye on the marshmallows, do not walk away. Rotate the tray as necessary to toast evenly and avoid burning. Remove the marshmallows at the desired toasted level then flip them over as best as possible and repeat toasting on the second side.
Once toasted, set aside to cool.
Begin warming the milk in a saucepan on the stove on medium low heat. When it begins to steam, whisk the sugar, eggs, and salt together in a separate medium bowl until a pale yellow.
Slowly stream the warmed milk into the egg/sugar mix while constantly whisking to temper, adding enough milk to warm the eggy mix.
Strain the egg/sugar/milk through a fine mesh strainer back into the saucepan and turn the heat up to medium. Whisk constantly until it thickens. Turn the heat to low and add the toasted marshmallows, stirring and lightly whisking until they mostly melt.
Remove the saucepan from the heat, add the cream and vanilla, stir and pour into a large bowl set in an ice bath or place the bowl in the refrigerator to cool.
Once the custard has chilled, churn in an ice cream maker following the manufacturer’s directions.
When the ice cream is ready, transfer to a freezer safe container and freeze.