Mix the flour and salt in a medium bowl.
Dissolve yeast in the warm milk and the sugar in a large bowl. Let sit for about 5–10 minutes until it looks foamy.
Add the egg, butter, and the flour mixture. Mix until a soft dough forms, adding more flour if needed.
Knead the dough on a floured surface for 6–8 minutes, until the texture is smooth and elastic. Place in a greased bowl, cover, and let rise about 1 hour or until doubled.
Preheat the oven to 350 degrees.
Punch down the dough and roll it into a 12x18-inch rectangle.
Spread the softened butter in an even layer on top of the dough.
Mix the apples, brown sugar, and cinnamon in a small bowl.
Spread the apple mixture over the buttered dough.
Roll up tightly from the long side and cut into 12 slices.
Place the rolls in a greased 9x13-inch pan.
Cover and let the rolls rise for 30–45 minutes, until puffy.
Bake for 25–30 minutes, until golden brown.
Whisk the glaze ingredients together in a small bowl and drizzle it over the warm rolls before serving.