Grease the inside of four 6-ounce glass or ceramic ramekins with butter or non-stick cooking spray. Set aside.
In a large mixing bowl, beat together the eggs, egg yolks, and sugar with a hand mixer on medium speed until foamy and light yellow in color.
In a small microwave-safe bowl, melt the chocolate chips and butter in the microwave on HIGH for 30 seconds. Stir and heat again. Repeat until melted and smooth. Whisk the melted chocolate into the egg mixture until incorporated.
Sift the flour and cocoa powder into the mixing bowl and whisk to combine. Add in the salt and the vanilla extract and whisk.
Divide the cake batter evenly between the prepared ramekins, and then refrigerate for 30 minutes.
Preheat the oven to 400 degrees F.
Place the ramekins into square casserole dish (8x8 or 9x9 should be big enough). Pour enough warm water into the casserole dish to come about halfway up the sides of the ramekins.
Bake for 10 to 15 minutes or until the tops of the cakes begin to look dry.
Remove the casserole dish from the oven and lift ramekins out of water. Let them cool on a wire rack 10 minutes.
Invert the ramekins onto a dessert plate. (You can run a knife around the edges of the cake to help it release easily if desired.) Dust the top of the cake with powdered sugar. Garnish with fresh raspberries before serving warm.