Step 1Heat the oven to 325 degrees.
Step 2Prepare the caramel: Place one-fourth cup of the sugar in a small, heavy-bottomed saucepan along with the corn syrup, adding enough water to give the sugar a "wet sand" consistency. Cook the sugar over medium heat until the sugar dissolves and the syrup darkens from a rich golden to a deep amber color, being careful that the caramel does not burn, 10 to 12 minutes. Remove from heat and, working quickly, divide the caramel among 6 (4-ounce) ramekins, coating the bottoms evenly. Set aside to cool and harden.
Step 3In a medium bowl, whisk together the remaining one-third cup of sugar and the eggs. Pour in the cream, half-and-half and vanilla, and whisk together gently so as not to incorporate air and over-mix. Set the mixture aside until the sugar dissolves. Strain mixture through a sieve into a pitcher or bowl with a lip. Set the caramel-coated ramekins in a roasting pan lined with a kitchen towel, and divide the custard evenly among the cups. Carefully fill the pan with hot water to reach a little more than the middle of the ramekins. Loosely cover the pan with foil and carefully place it in the oven and bake for 25 minutes. Rotate the pan and cook until the custard is set but the center still jiggles when the dish is nudged, 20 to 25 more minutes.
Step 4Remove the pan from the oven, uncover and set aside to cool in the pan for about 30 minutes. Transfer the ramekins to a plate to cool completely, then cover with plastic wrap and refrigerate overnight.
Step 5Remove the custards from the refrigerator and let them sit at room temperature for 10 minutes before serving. To serve, gently run a thin knife around the inside of the ramekin to loosen the flan. Cover with a plate and invert the ramekin onto the plate and allow the custard to come loose onto the plate. The caramel will pool around the flan (if the custards are stuck, gently run the base of each ramekin over a low stove-top flame to warm the caramel slightly and make it easier to dislodge the custards). Garnish with orange segments, whipped cream and fresh mint, if you like.