Pre-heat oven to 350. Grease 5-6 small ramekins or 4-5 medium ramekins and place in a cake pan filled about halfway full with room temperature water
In a sauce pan, combine butter and roughly chopped chocolate over low heat, stirring constantly. When about half of the chocolate has melted, add in the vanilla extract and turn off the heat. Continue stirring until all of the chocolate and butter have melted and the mixture is smooth and well-combined. Set aside to cool.
In a medium-sized bowl, or in the bowl of a stand mixer, add the room temperature eggs and sugar. Whisk for about 1 minute, until whites and yolks are combined with the sugar.
When the chocolate is still very warm, but not hot, slowly stream the chocolate and butter mixture into the eggs, mixing constantly while pouring. I like to pour the mixture down the side of the bowl so it further cools before mixing with the eggs.
Once you've added all of the chocolate, slowly add in the flour while mixture is on until all flour has been added. If there is any separation in your batter or if it looks greasy, add 1 tablespoon of flour and stir until batter is cohesive. Repeat as necessary. This usually happens if your eggs are too cool and not room temperature. The batter should be glossy and smooth.
Divide batter evenly among your greased ramekins that are inside your cake pan, that has been filled with an 2-3" or so of water, or until water is no more than about halfway up the ramekins.
Bake time will depend on how gooey you want your lava brownies OR if you plan to reheat them (they are really good reheated!). If you like a very gooey brownie/you're planning to reheat them and you're using a small ramekin, you want to bake them for about 25 minutes. For the same doneness in a medium ramekin, you'll want to let them bake for about 30 minutes. If you want a mostly-done brownie, you could let them bake for 35-40 minutes. You'll just want to use a toothpick to check for desired doneness :) They are done when the tops are shiny, the edges will be set and the center will be gooey. Let cool for 5 minutes and enjoy!