Preheat the oven to 450°F. Toss together the pineapple and brown sugar and spread in an even layer on a large rimmed baking sheet. Roast for about 15 minutes, until tender and lightly caramelized. Remove from the oven and let cool completely.
Pour the cream into a large bowl and place over an ice bath. Place a fine mesh sieve over the bowl.
Whisk the egg yolks in a large bowl. Meanwhile, combine the milk, sugar, and salt in a medium saucepan over medium-high heat. Bring the mixture to a gentle simmer, then remove from the heat. Slowly add the warm milk mixture to the egg yolks, whisking until well combined. Transfer the mixture back to the saucepan and cook over medium-low heat for 5 to 10 minutes, stirring constantly, until the mixture is thick enough to coat the back of a wooden spoon. Strain the custard through the sieve and into the heavy cream. Stir for about 5 minutes, until the mixture is cool.
Place the pineapple in the bowl of a food processor and pulse until smooth. Stir the pineapple and vanilla extract into the ice cream custard. Cover the bowl with plastic wrap and chill in the refrigerator for at least 4 hours, or overnight.
Pour the custard into an ice cream maker and process according to the manufacturers’ instructions. Layer the ice cream in an airtight container, alternating the ice cream with the caramel. Freeze for about 3 hours, until firm.