Prepare a small loaf pan with parchment, greasing the sides. Preheat the oven to 350°F/180°C.
In a small saucepan at medium heat, cook half the cranberries with ¼ cup of water until the cranberries have bursted, reduced and become a thick puree that resembles apple sauce. This will take 10-15 minutes. Allow cranberry puree to cool down. Then add the baking soda and incorporate to the puree. Reserve.
In a medium sized bowl mix the flour, baking powder and salt. Reserve.
In the bowl of a standing mixer with the paddle attachment add the sugar, unsalted butter, egg, lemon zest, and vanilla extract. Whip at medium high speed for 2-3 minutes until the mixture is clear and creamy. Add the Greek yogurt and cranberry puree and mix until combined. Then, add the dry ingredients and mix until the dough is incorporated.
Take the bowl off the standing mixer and add the second half of cranberries. Mix with a spatula until they are equally distributed in the cake mix.
Add the cake mixture into the prepared loaf pan and create a flat surface with the back of a spoon. Bake for 40-50 minutes, until the cake is golden brown and a toothpick comes out clean when inserted. Allow to cool completely.
To make the frosting, blend 2 tablespoons of chopped cranberries and 2 tablespoons of lemon juice and sieve the resulting liquid. Mix the cranberry juice with powdered sugar. Glaze the cooled cake and garnish with lemon slices.