In a small bowl, combine the warm milk, yeast, and 1 tablespoon sugar. Let stand for 5 to 10 minutes until foamy.

In a large bowl, whisk together 2 1/2 cups flour, 1/4 cup sugar, and salt.

Add the eggs, softened butter, sour cream, vanilla, and yeast mixture. Mix until a soft dough forms.

Knead the dough on a lightly floured surface for 6 to 8 minutes, or until smooth and elastic. Place in a greased bowl, cover, and let rise in a warm place for 1 to 1 1/2 hours, until doubled in size.

Grease a 9x13-inch baking pan or a similar large baking dish.

In a separate bowl, toss the fruit with 1/4 cup sugar, cinnamon, 2 tablespoons flour, and lemon juice until evenly coated.

Punch down the risen dough and press it evenly into the prepared pan. If needed, lightly flour your hands to make it easier to spread.

Arrange the fruit mixture evenly over the dough. Drizzle the melted butter over the top and sprinkle with coarse sugar if using.

Cover loosely and let rise again for 20 to 30 minutes while you preheat the oven to 350°F (175°C).

Bake for 35 to 45 minutes, or until the kuchen is golden brown and the fruit is bubbling.

Let cool for at least 15 minutes before slicing and serving warm.