Warm the milk to 105–115°F. In a large bowl, whisk the warm milk, granulated sugar, and instant yeast; let stand until foamy, 5–10 minutes.
Add the egg, softened butter, kosher salt, and 2.5 cups of the all-purpose flour. Mix until a shaggy dough forms, then add the remaining flour as needed to make a soft, slightly tacky dough.
Knead until smooth and elastic, 8–10 minutes (stand mixer on medium speed or by hand).
Lightly grease a clean bowl and a 9×5-inch loaf pan with the vegetable oil. Shape the dough into a ball, place in the greased bowl, turn to coat, cover, and let rise until doubled in size, 60–90 minutes.
For the filling, stir together the brown sugar, ground cinnamon, and the 1 tbsp all-purpose flour until evenly combined.
Gently deflate the dough and turn it out onto a clean surface. Roll into a 9×18-inch rectangle with the short side 9 inches.
Beat the remaining egg to make an egg wash. Brush a thin, even coat over the dough, leaving a 0.5-inch border unbrushed on all sides. Sprinkle the cinnamon-sugar filling evenly over the egg-washed surface and gently press to adhere.
Starting from a 9-inch short edge, roll the dough up tightly into a log. Pinch the seam and the ends to seal well, and set the log seam-side down in the prepared pan.
Cover and let rise until the loaf crowns about 1 inch above the rim and springs back slowly when poked, 45–60 minutes. Meanwhile, heat the oven to 350°F (175°C).
Brush the top lightly with any remaining egg wash. Bake until deep golden brown and an instant-read thermometer inserted in the center reads 190–195°F, 40–45 minutes; tent with foil after about 25 minutes if the top is browning too fast.
Cool in the pan for 10 minutes, then turn out onto a rack and cool completely, at least 1 hour, before slicing.