In a large saucepan, heat the butter, both sugars, water, corn syrup, lemon juice, and salt over medium-low heat. Slowly melt the mixture. Continue heating the mixture, stirring frequently. Eventually, the mixture will bubble and become stringy. Continue cooking until it reaches 300°f (~15-20 minutes).
Remove saucepan from the burner and pour in 1/4 cup cream into caramel, stirring well. It will bubble vigorously; don't be alarmed. Whisk the mixture until smooth.
Return saucepan to medium heat and slowly whisk in 3/4 cup cream and 2 cups milk.
Warm 1/4 cup milk in microwave for ~30 seconds. Whisk together the egg yolks and cornstarch in large heat-safe bowl, then slowly whisk in the warm milk.
Check on caramel mixture. Gradually increase heat. As soon as the mixture comes to a full boil and starts climbing the walls of the saucepan, pour all of the caramel mixture into the egg yolk, cornstarch, and milk mixture. Whisk vigorously until thickened. Add vanilla and rum, and continue whisking.
Cover with parchment paper, in order to prevent a layer of skin from developing. Refrigerate for at least 3 hours. Enjoy!