In a large bowl, mix together all dry ingredients - cornmeal, flour, salt and baking powder.
In a separate bowl, add melted butter and sugar, mix well until sugar is dissoved.
Add the yogurt (or applesauce) and mix well to combine with the butter sugar mixture. It is a good idea to whisk the yogurt well before adding to the butter mixture, so there are no lumps and it is easy to mix.
Also add the milk and vanilla extract to the butter mixture and mix everything together until well combined.
Now add the prepared butter mixture to the dry flour mixture and mix the contents together without forming any lumps. In case the batter is too thick (as it may depend on the cornmeal and flour you use), you may add a little more milk and mix again( just add one tablespoon of extra milk at a time and then mix and see if you need more). You can use a balloon whisk to mix the ingredients to blend in perfectly.
Preheat the oven in 350 degree Fahrenheit.
Grease oil or melted butter in a square pan and pour the cornbread batter into it.
Bake the cornbread for about 35 to 40 minutes until the top is lightly browned and the inside is cooked. You can check this by inserting a toothpick at the center of the cornbread, which should come out clean. If the inside is not cooked enough, cover the pan with a foil and cook for few more minutes until the cornbread is cooked completely. Covering with the foil helps to prevent the cornbread from drying out.
Allow the bread to cool down, then cut and serve the warm cornbread with tea, coffee or milk along with honey or butter or jam or any syrup as you wish.