Strawberry Swirl Buns

milk*

cream cheese, at room temperature

unsalted butter, at room temperature

granulated sugar

large egg

instant yeast

salt

unbleached all-purpose flour

fresh strawberries, hulled and quartered

granulated sugar, divided

cornstarch

lemon juice

water

cream cheese, at room temperature

confectioners’ sugar, sifted

freshly squeezed lemon juice

milk or cream, plus more as needed