Warm milk to about 100–110°F and add to the mixer bowl with 1 packet instant yeast, 2 tablespoons sugar, 3 cups flour, and 1 teaspoon salt; add the beaten egg and 3 tablespoons softened butter and mix with a dough hook on low until combined, then knead 5–7 minutes until smooth and slightly tacky; add up to 1/4 cup flour only if very sticky.

Place dough in a lightly oiled bowl, cover, and let rise until doubled, about 60–90 minutes at room temperature.

In a small saucepan combine 2 cups blueberries, 1/3 cup sugar, 1 tablespoon lemon juice, 1/2 teaspoon cinnamon, and 1 tablespoon water; simmer until berries begin to soften, then whisk 1 teaspoon cornstarch mixed with 1 teaspoon cold water into the pan and cook 1–2 minutes until glossy; cool to warm or room temperature.

Roll risen dough into a 12×16-inch rectangle on a lightly floured surface; spread 3 tablespoons softened butter, sprinkle with brown sugar mixed with 1 tablespoon cinnamon, then spoon blueberry filling evenly over the sugar, leaving a 1/2-inch border.

Roll the dough tightly from the long side into a log, pinch the seam to seal, and tuck the ends under; chill the log in the refrigerator 20–30 minutes to firm for slicing.

Preheat oven to 375°F. Slice the chilled log into 12 equal rolls using a serrated knife or dental floss and place them in a greased 9×13-inch pan in three rows of four; cover loosely and let rise 30–45 minutes until puffy.

Bake at 375°F for 18–22 minutes until golden and centers are set (about 190°F internal if measured); tent with foil if tops brown too fast.

Whisk 1 cup powdered sugar with 2–3 tablespoons milk and 1/2 teaspoon vanilla until smooth; drizzle over warm rolls and serve. Cool remaining rolls before storing.