In a mixing bowl combine together greek yogurt and icing sugar until the sugar is completely dissolved. Even after beating it for a while if the sugar does not dissolve (check by rubbing it in between your fingers for no signs of grittiness) simply cover it with a cling wrap and set it aside on your counter top.
Meanwhile, chop your chocolates and place it in a separate bowl. Add butter to it.
Using the double boiler method, melt the chocolate and butter until homogenous. Set this aside too.
Weigh out the dry ingredients - flour, cocoa powder, baking soda and salt in a small bowl. Set it aside.
Check on your yogurt sugar mixture, by now the sugar would have dissolved completely. To this add the melted chocolate mixture and mix well.
Pass the dry ingredients through a sieve directly on top of the wet ingredients. (Sieving would help remove lumps from cocoa powder and baking soda).
Using a spatula combine to form a thick brownie batter.
Transfer it to the prepared square pan and bake it in the preheated oven of 160°C/320°F for 30 to 35 mins or until a skewer inserted in the centre comes out with few crumbs sticking to it.
Let it sit in the pan itself for another 5-10 mins and then lift it using the hanging parchment paper and transfer to a wire rack to cool completely for a minimum of 1-2 hours before cutting into thick squares.
It is always a better idea to start preparing your cake pan by applying a thin layer of butter or oil and then placing a parchment paper hanging out on to the edges of the pan so that it is easier to lift the brownies once baked.
Ensure that the sugar is completely dissolved in the yogurt by rubbing it between your clean fingers. You should not feel any grittiness.
The recipe would work with compound chocolate too. But using couverture chocolate would make it the ultimate eggless brownies recipe in terms of enhancing the flavour.
Sieve the dry ingredients to remove the lumps from cocoa powder and baking soda.
The consistency of brownie batter should be thick. To achieve that make sure you are using greek yogurt. If not then hang your yogurt for couple of hours to separate the whey. Homemade yogurt will not work for this recipe. Buy store bought and hang it for couple of hours if you cannot find greek yogurt. Please note you might want to start with 200-220 gms of yogurt if making hung curd and then use 100 gms of thick part of yogurt to make the recipe.
Other general tips like preheating your oven before baking and having all your ingredients at room temperature applies to this as well.