In a small saucepan combine the flour and milk for the tangzhong. Cook over low heat, whisking continuously, until the mixture thickens to a mashed potato consistency, about 4-5 minutes. Remove from heat and transfer to a small bowl to cool.
In the bowl of a stand mixer combine bread flour, sugar, and salt. Add summer squash, melted butter, sourdough starter, and egg. Add tangzhong once it has cooled. Mix together with a rubber spatula or wooden spoon until a shaggy dough forms.
With a dough hook attachment, knead dough on medium-high speed for about 8-9 minutes or until dough is smooth and tacky. Perform a window pane test to evaluate gluten development (see notes).
Transfer dough to a clear bowl or container that has been lightly greased with oil. Cover loosely with plastic wrap or a kitchen towel. Leave at room temperature until doubled in volume, about 4-6 hours (see notes).
Butter a 9x5-inch loaf pan. Punch down dough and transfer to a floured surface. Knead briefly to knock out excess air. Divide dough into 8 equal pieces (use a scale if you have one for accuracy). Pinch and pull the sides of each piece of dough into the middle to form them into taut balls.
Place dough balls seem-side down, side-by-side in the loaf pan.
Cover and allow dough to rise until about doubled. This may take around 3-4 hours.
When ready to bake, preheat oven to 350°. In a small bowl whisk together egg and heavy cream for egg wash. Brush the top of the loaf with egg wash.
Bake 40-50 minutes until golden brown and internal temperature of 190°. If needed, cover the sides with foil for the last 10 minutes of baking if they are browning too quickly. Remove to a wire rack to cool completely before slicing.