Almost-Famous Molten Chocolate Cake

INGREDIENTS

• 1/2 teaspoon salt
• 1/2 cup heavy cream
• 2 large eggs, at room temperature
• 1/2 teaspoon baking powder
• Caramel sauce, for drizzling
• 1 1/3 cups all-purpose flour
• 1 pint vanilla ice cream
• 8 ounces bittersweet chocolate, finely chopped
• 1 1/2 teaspoons vanilla extract
• 1 1/3 cups sugar
• 1 teaspoon baking soda
• 1 tablespoon light corn syrup
• 3 tablespoons milk
• 1/2 cup natural (not Dutch-process) cocoa powder, plus more for dusting
• 4 tablespoons unsalted butter
• 1/4 cup vegetable oil
• 6 tablespoons unsalted butter (2 tablespoons melted, 4 tablespoons at room temperature)