Preheat the oven to 300°F. Line a large rimmed baking sheet with parchment paper or a silicone baking mat.
Toss the strawberries with 1/4 cup sugar and the balsamic vinegar. Spread strawberries into one even layer on the prepared baking sheet and roast for about 30 minutes, or until the strawberries have darkened in color, are soft, and their juices have released. Transfer the strawberries to a medium bowl and mash until only small pieces of fruit remain. Cover and refrigerate the strawberries until chilled about 1 hour.
Prepare an ice bath by filling a large bowl with ice cubes and a cup or two of water. Place a medium bowl fitted with a fine strainer inside ice bath.
In a medium saucepan combine the remaining 1/2 cup sugar, with the milk, cream, salt, and vanilla. Set over medium heat, stirring occasionally, until the mixture is warm and begins to steam, about 5 minutes.
Whisk together the egg yolks until well-combined and pale in color. Carefully whisk half of the warm milk mixture into the egg yolks, until the egg mixture is warmed. Whisk the egg-milk mixture back into the saucepan. Cook the mixture over medium heat, stirring constantly, until the mixture is thick enough to coat the back of the spoon and registers around 175°F on an instant-read thermometer, about 5 to 7 minutes. Be careful not to boil the mixture.
Immediately strain the mixture through the fine strainer into the prepared ice bath. Cool the custard in the ice bath until it’s at room temperature, stirring often. Press plastic wrap against the surface of the custard and refrigerate until chilled, about 4 hours or up to 1 day.
Pour the custard into an ice cream maker. Freeze according to the manufacturer’s directions. During the 5 last minutes of freezing, gradually add the roasted strawberries. During the last minute of freezing add the vodka. Transfer the ice cream to an airtight container, press plastic wrap against the ice cream, and freeze until firm and flavor is ripened, at least 2 hours.