Place a fine-mesh sieve over a medium bowl set in a large bowl filled 1/3 of the way with ice water and set aside
Sprinkle a thin even layer of sugar into large heavy saucepan. Cook undisturbed over medium heat until mostly melted, about 4 minutes
Sprinkle another layer of sugar over and cook, stirring around edges with a heatproof spatula to move melted sugar to center, until mostly melted, about another minute or so
Repeat until you've used 1.25 cups of sugar and it's mostly melted, about 6-8 minutes
Continue cooking, swirling saucepan until caramel is very flui and turned dark amber, about 5 minutes
Remove from heat and add heavy cream - caramel will seize, but that's ok. Return pan to low heat and stir until hardened caramel is dissolved
Stir in vanilla and salt; set aside
In a medium bowl, whisk egg yolks and 2 tbsp sugar until lightened and thick, about 1 minute. Whisking constantly, add 1/2 cup caramel gradually to egg yolk mixture. Whisk mixture back into saucepan and cook over medium-low heat, whisking constantly, until custard is thick enough to coat a spoon and just holds the marks of the whisk, about 5 minutes
Immediately pour through sieve into bowl. Stir constantly until cold, about 5 minutes. Whisk in milk. If not super cold, refrigerate for a half hour at this stage
Churn in ice cream maker according to directions. Freeze until firm - at least 4 hours