In a large mixing bowl, combine the warm milk, 1/4 cup granulated sugar, and yeast, whisk to dissolve, and let sit for 5–10 minutes until foamy.
Add the eggs, melted butter, and vanilla bean paste to the yeast mixture and whisk until smooth.
Add 3 1/2 cups of the flour and the salt to the bowl and stir with a wooden spoon or dough hook until a soft, shaggy dough forms.
Gradually add the remaining 1/2 cup flour, a little at a time, mixing until the dough is soft, slightly tacky, and pulls away from the sides of the bowl.
Turn the dough out onto a lightly floured surface and knead for 8–10 minutes, until smooth, elastic, and stretchy.
Lightly grease a large bowl with about 1 tablespoon of the softened butter, place the dough in the bowl, turn to coat, cover with a clean towel, and let rise in a warm place for about 1 hour, or until doubled in size.
While the dough rises, make the filling by beating the cream cheese, remaining 1/4 cup granulated sugar, and vanilla bean paste in a medium bowl until smooth and creamy.
Gently fold the diced strawberries into the cream cheese mixture until evenly distributed and thick and spreadable.
Lightly grease a 9x13-inch baking dish with the remaining softened butter and set aside.
Once the dough has doubled, punch it down gently to release air and transfer it to a lightly floured surface.
Roll the dough into a large rectangle about 12x18 inches, with the long side facing you.
Spread the strawberry cream cheese filling evenly over the dough, leaving a small 1/2-inch border along the long edge farthest from you.
Starting from the long edge closest to you, roll the dough tightly into a log, pinching the seam to seal once fully rolled.
Trim the ends if needed, then use a sharp knife or dental floss to cut the log into 12 equal rolls.
Arrange the rolls, cut side up, in the prepared baking dish, spacing them evenly.
Cover the baking dish with a towel and let the rolls rise in a warm place for 30–45 minutes, until puffy and almost doubled.
Near the end of the rising time, preheat the oven to 350°F and position a rack in the center of the oven.
Bake the rolls for 25–30 minutes, until golden brown on top and the centers of the rolls are cooked through.
Remove the pan from the oven and let the rolls cool in the pan on a wire rack for 10–15 minutes.
For a light icing, whisk the powdered sugar with 1 tablespoon of milk or cream, adding more milk a little at a time until smooth and pourable.
Drizzle the icing over the warm rolls or dust generously with powdered sugar, then serve slightly warm.