Preheat the oven to 325° F. Set rack in the middle of the oven.
In the bowl of the stand mixer, cream the butter and sugar until it it light and fluffy.
Add the eggs, one at a time, beating well each time. Add the applesauce and vanilla extract, and mix well.
In the medium bowl, sift together the flour, cinnamon, nutmeg, and baking soda. Add to the butter mixture and stir gently, tossing in the cranberries. Do not overmix.
Pour the batter into the prepared pan. Bake for 60 to 75 minutes, checking every 5 minutes after 1 hour has passed. When a tester comes out clean, remove the pan from the oven.
Cool the cake in the pan for 15 minutes, then turn the cake out onto a wire rack and cool completely.
While the cake is cooling, combine the lemon juice, orange juice, and lavender buds in a small saucepan. Bring to boiling over medium heat. Remove from heat and strain into small bowl. Allow to cool. Add the powdered sugar and whisk until the glaze is smooth.