Whisk the egg yolks in a large heatproof bowl and set aside.
Combine the granulated sugar and water in a small saucepan over high heat, stirring until the sugar is dissolved. Bring to a boil and cook for 5 to 6 minutes, until the mixture registers 325°F on a candy thermometer and turns a light golden color. Remove from the heat and let cool for 1 minute. Stir in ¼ cup of cream, then set aside and let cool completely.
Whisk together the milk, brown sugar, cornstarch, and salt in a medium saucepan over medium-high heat and cook until the mixture comes to a gentle boil. Remove from the heat and whisk in the cooled caramel. Slowly add the warm milk mixture to the egg yolks, whisking vigorously until well combined. Return the mixture to the saucepan and place over medium-low heat. Cook for 5 to 8 minutes, stirring constantly, until the mixture is thick enough to coat the back of a wooden spoon. Remove from the heat and stir in the butter and vanilla extract.
Spoon the pudding into six 4-ounce ramekins and place on a large rimmed baking sheet. Cover with plastic wrap and chill in the refrigerator for at least 3 hours, until cold. Spoon the salted caramel sauce over the tops of the puddings and sprinkle with sea salt. Chill for another hour, until firm.
Place the remaining ¼ cup of cream in the bowl of an electric mixer fitted with the whisk attachment. Beat on medium-high speed until stiff peaks form. Top each pudding with a dollop of whipped cream.