Place 2 tablespoons unsalted butter in a small bowl and let sit out at room temperature to soften while you prepare the dough.
Place 1/4 cup plus 2 tablespoons water and 2 tablespoons of the bread flour in a small saucepan and whisk until smooth. Heat over low heat, stirring constantly with a rubber spatula, until the mixture becomes thick and paste-like, 1 to 2 minutes. Immediately scrape into a large bowl and let cool to room temperature.
Place 1/2 cup whole milk in a small microwave-safe bowl or measuring cup. Microwave on high until warm but not hot (about 100°F), 20 to 30 seconds. Sprinkle 1 1/2 teaspoons active dry yeast over the milk, stir to dissolve, and set aside until small bubbles form around the edge of the bowl, 5 to 7 minutes. Meanwhile, whisk together the remaining 2 1/2 cups bread flour, 1 1/2 teaspoons kosher salt, and 2 tablespoons granulated sugar together in a medium bowl.
Add the yeast mixture, 1/3 cup heavy cream, and 1 of the large eggs to the cooled flour paste and whisk until smooth. Add the flour mixture and stir with a wooden spoon until a shaggy dough forms.
Transfer the dough onto a lightly floured work surface and knead until a rough ball of dough forms, 2 to 3 minutes. If the dough is too sticky, knead in 1 tablespoon flour at a time until it comes together. Add the softened butter and knead until the dough is completely smooth, 3 to 4 minutes, adding more flour if needed to prevent sticking.
Coat a large bowl with butter, then transfer the dough into it. Cover with a kitchen towel or plastic wrap and let rise in a warm place until doubled in volume, 1 1/2 to 2 hours. Meanwhile, line an 8 1/2-inch loaf pan with parchment paper so that there is excess parchment hanging off the long sides, then coat the parchment and sides of the pan with butter.
Punch down the dough and transfer it onto a work surface. Divide the dough into 4 pieces. Using a rolling pin, roll one piece of dough into an 8×5-inch rectangle. Fold a third of the dough lengthwise over toward the center, then do the same thing from the other side. Starting at a short end, roll the dough up into a tight coil. Repeat with the remaining dough pieces.
Place the coils seam-side down in the loaf pan with the spirals against the long side of the pan. Let rise uncovered in a warm place until doubled in volume, 30 to 40 minutes. Meanwhile, arrange a rack in the middle of the oven and heat the oven to 350°F. Crack the remaining egg into a small bowl and whisk until no streaks remain.
Brush the beaten egg onto the dough and sprinkle with flaky salt if desired. Bake until the top is golden-brown and shiny, 25 to 30 minutes. Let cool for at least 30 minutes before slicing.