Preheat the oven to 350°F. Grease a 10 X 2-inch round cake pan with 1 teaspoon of the butter. In a medium-size nonreactive saucepan, combine the raspberries, 1/2 cup of the granulated sugar, and the lemon juice. Bring the mixture to a boil and cook at a boil for 3 minutes. In a small bowl, dissolve the cornstarch in the water. Add to the raspberry mixture and cook, stirring, until it thickens, about for 2 minutes. Remove from the heat and let cool completely.
In a large mixing bowl, cream 1/4 pound (1 stick) of the butter with the remaining 1 cup granulated sugar with an electric mixer. Add the eggs one at a time, beating well after each addition. Into a medium-size mixing bowl, sift 3 1/2 cups of the flour, the baking powder, baking soda, 1/2 teaspoon of the cinnamon, and the salt. Add the flour mixture, buttermilk, and vanilla to the butter mixture and beat with the electric mixer until everything is incorporated.
In another medium-size mixing bowl, combine the remaining 1/4 cup (1/2 stick) butter, the remaining 1/2 cup flour, the brown sugar, and the remaining 1/4 teaspoon cinnamon. Using your hands, combine the mixture until it resembles fine crumbs.
Spread half of the batter in the prepared pan. Spread the raspberry mixture over the batter. Drop heaping spoonfuls of the remaining batter over the raspberry mixture, about 1 inch apart. Sprinkle the crumb mixture evenly over the surface. Put the cake pan on a baking sheet and bake until golden brown, about 40 minutes. Remove from the oven and let cool slightly.
In a medium-size mixing bowl, whisk together the confectioners' sugar, cane syrup, and milk until smooth. Drizzle the frosting over the top of the coffee cake. Cut into slices and serve warm.
Yield: one 10-inch cake; 12 servings