Preheat oven to 325 degrees F. Place six 6-ounce custard cups at least 1/2-inch apart in a 3-inch deep baking pan; set aside.
To caramelize sugar, place 1/2 cup of the sugar in a large skillet. Place skillet over medium-high heat and cook until sugar starts to melt, shaking skillet occasionally. Do not stir. When sugar starts to melt, reduce heat to low and cook about 5 minutes or until all of the sugar is melted, stirring as needed with a wooden spoon. Working quickly, pour syrup into custard cups, dividing syrup evenly among cups. Tilt cups to distribute caramel evenly over the bottoms of the cups. Set aside.
In a medium saucepan combine the remaining 1/4 cup of sugar, the milk, and the condensed milk. Cook and stir over medium heat until mixture is simmering. Remove from heat.
In a large bowl whisk eggs and egg yolks until combined. Gradually add milk mixture, whisking well. Stir in vanilla. Strain mixture through a fine mesh sieve into a pitcher or large liquid measuring cup. Divide mixture among custard dishes.
Pull out oven rack; set pan with custard cups on rack. Using a tea kettle, slowly pour hot water into the baking pan around the cups until the water reaches about halfway up the sides of the custard cups. Slowly and carefully slide oven rack back into the oven.
Bake flans for 65 to 70 minutes or until barely set in the centers. Remove pan from oven; transfer to a cooling rack. Let flans cool in the hot water for about 1 hour. Remove flans from water. Cover and chill for at least 2 hours.
To serve, run a small knife around the edges of one flan. Quickly turn custard cup over a serving plate. Shake plate and custard cup to release the flan. Remove custard cup. Using a small rubber spatula, scrape excess caramel from the cup over the flan. Repeat with remaining flans. Sprinkle each flan with salt. Spoon blackberries over flans; serve immediately.