Heat the oven to 350°F.
Place 3 pounds peeled and cut sweet potatoes in a large pot and add enough cold water to cover by about 1 inch. Bring to a boil over high heat. Reduce the heat to maintain a simmer and cook until fork-tender, 12 to 15 minutes.
Drain the potatoes, then return to the pot. Add ½ cup packed brown sugar, ½ cup whole milk, 6 tablespoons room-temperature unsalted butter, 1 teaspoon vanilla extract, and ¾ teaspoon kosher salt. Mash with a potato masher until smooth.
Transfer to a 9x13-inch baking dish and spread into an even layer. Arrange 2 ½ cups miniature marshmallows evenly over the top in a single layer. Bake until the sweet potatoes are heated through and the marshmallows are golden-brown and puffed, 18 to 20 minutes.
