Cinnamon-Sugar Coffee Cake

Ingredients

• 1/2 cup walnuts, chopped
• 1/4 cup dark brown sugar, packed
• 2 1/4 cups superfine sugar
• 1 1/2 sticks unsalted butter, very cold
• 1/2 teaspoon ground cinnamon
• 1/2 teaspoon, plus 1/4 teaspoon ground ginger
• 1/2 teaspoon kosher salt
• 1/4 teaspoon baking soda
• 1 cup milk
• 1/4 cup fresh lemon juice
• 1 tablespoon heavy cream
• 1 teaspoon vanilla extract
• 1/3 cup sliced almonds or chopped walnuts
• 2 tablespoons candied ginger, minced
• 3 eggs
• 3 egg yolks
• 1 cup confectioners' sugar, sifted, plus extra, as needed
• 1 to 2 tablespoons milk
• 1/2 teaspoon ground cinnamon
• 1/2 teaspoon vanilla extract
• pinch of salt
• pinch of ground allspice (optional)