Beat the butter and sugar on medium speed in a heavy-duty electric mixer fitted with the paddle attachment until fluffy, about 3 minutes. Stir in the egg.
Switch to the dough hook. Add the flour, yeast, salt and milk and mix on medium speed until the dough is smooth 3-5 minutes. It will feel slightly sticky, Cover the bowl and let the dough rise in a warm spot until doubled in size, 45 minutes - 1 hour.
Make the filling:
While the dough rises, put the strawberries, sugar, water and cornstarch in a small saucepan over medium heat. Bring to a simmer, stirring frequently, until the strawberries begin to soften and the juices thicken, about 5 minutes. Remove from the heat and transfer to a bowl to cool completely.
Make the sweet rolls
Brush a 9 x 13-inch baking pan with the melted butter.
Gently deflate the dough and coax it out of the bowl onto a lightly floured surface. Roll out the dough into a 12 x 18-inch rectangle with a rolling pin, dusting with a little more flour if it feels sticky.
Spread the filling evenly over the dough, leaving a 1/2-inch border around the edges. With the short side facing you, roll up the dough away from you to form a log. Roll as firmly as possible, taking care not to squeeze out the filling. Pinch the dough at either ends to seal and turn the roll seam side down. Slice the dough with scissors or a sharp knife into 12 1-inch wide pieces.
Place the rolls cut side up in the prepared baking pan. Cover and let rise until they're doubled in size, about 45 minutes. Alternatively, cover with plastic wrap and refrigerate overnight.
Preheat the oven to 350 degrees. Bake the rolls until light golden brown, 20-25 minutes. Cool on a rack for 10 minutes while you make the glaze.
Make the glaze:
Whisk all ingredients in a small bowl until smooth.
To serve
Drizzle the glaze over the top of the rolls, spreading with a spatula. Serve warm or at room temperature.
Notes
For morning buns: Make the dough recipe through step 4 (cut and arranged in baking pan) and refrigerate overnight. In the morning, let the buns rest at room temperature while the oven preheats. Bake and glaze following the recipe instructions.
Leftover rolls will keep up to 2 days at room temperature.
To make the dough by hand, blend the butter and sugar in a large bowl with a wooden spoon or use a hand mixer. Mix in the egg, flour, yeast, salt and milk until a soft dough forms. Turn it out onto a lightly-floured surface and knead by hand until it forms a soft, smooth ball. Add enough flour to keep the dough manageable, dusting with up to 1/4 additional flour if needed, then proceed with the rising step.