Preheat the oven to 350°F/180°C.
Melt the butter in a medium skillet over medium heat. Add garlic and sauté for 1 to 2 minutes until fragrant. Stir in tomato paste, paprika, salt, and basil. Pour in the boiling water, cover, and simmer for 3 minutes. Add Parmesan and cook for 1 minute more. Pour the sauce into a 9x5-inch/23x12 cm loaf pan.
Mix the feta cheese, mozzarella, cream cheese, egg white, breadcrumbs, basil, and salt in a bowl until smooth.
Using a vegetable peeler, slice the zucchinis lengthwise into thin ribbons. Place 1 teaspoon of cheese filling at the end of each slice and roll tightly. Arrange the rolls upright in the prepared pan with the sauce.
Bake uncovered for 45 minutes, until the zucchini is tender but still holds its shape. Let rest for 15 minutes before serving to allow the sauce to thicken. Serve warm.