Heat the milk and butter in a small saucepan over medium-low heat just until small bubbles start to form around the edges of the pan and the butter is mostly melted. Remove from the heat and set aside to cool until lukewarm.
3/4 cup whole milk, 1/2 cup salted butter
In a small bowl, combine the warm water, yeast, and 1 teaspoon of the sugar. Let it proof until foamy, about 5 minutes. You can skip this step if using instant yeast, but it's necessary for active dry yeast and I have found that I get a better rise even with instant yeast if I let it proof first.
1/2 cup warm water, 2 1/4 teaspoons instant yeast, 1/4 cup (50g) granulated sugar
In a large bowl, combine the proofed yeast, remaining sugar, egg, 1 cup of the flour, salt, and the cooled milk mixture. Mix well until smooth and combined using the paddle attachment of your stand mixer.
1 large egg, 3 3/4 cups (529g) all-purpose flour, 1/2 teaspoon salt
Add the remaining flour and switch to the dough hook attachment. Knead for 5-7 minutes until the dough starts to pull off from the sides and is nice and smooth. It should be soft and tacky, but not too sticky. Cover loosely with plastic wrap or a clean cloth and let it rise in a warm spot for 1 to 1 1/2 hours or until almost doubled in size.
Once the dough has risen, turn it out onto a clean floured surface and roll it out until about 1/4 to 1/2-inch thick until you have a roughly 9x12-inch rectangle.
Gently spread 5 tablespoons of softened butter evenly over the dough, then sprinkle with the brown sugar. Toss the raspberries in a bowl with the granulated sugar, lemon zest, and cornstarch until evenly coated. Sprinkle evenly over the dough.
5 Tablespoons salted butter, 1/3 cup light brown sugar, 1/3 cup granulated sugar, 18 ounces raspberries, 2 Tablespoons lemon zest, 1 1/2 teaspoons cornstarch
Use a pizza cutter to slice the dough into 12 strips, then roll up each roll individually, tucking in any raspberries that try to escape. Transfer the shaped rolls to a greased 9x13-inch baking pan. You can roll the dough up as a long, then slice it into rolls with a sharp knife or using the thread or floss method, but I have found that it's easier for me to use the other approach of slicing them first.
Cover with plastic wrap and let it rise for 30 minutes or refrigerate overnight.
While the rolls are rising, preheat the oven to 375°F.
Bake for 25 to 30 minutes until golden brown and done in the middle. If the rolls are browning too much on top, cover them loosely with a piece of aluminum foil. The internal temperature of the rolls should be between 190 and 200°F when tested with a digital thermometer.
In a large mixing bowl, beat the softened butter and cream cheese with a hand mixer until creamy, smooth, and completely combined. Add powdered sugar, sour cream, mashed raspberries, vanilla, and lemon zest, then mix again until spreadable and combined. Spread the icing over the rolls while still fresh out of the oven. Let sit for 5-10 minutes before serving.
3 Tablespoons salted butter, 4 ounces cream cheese, 1 cup powdered sugar, 2 Tablespoons sour cream, 1 teaspoon pure vanilla extract, Zest of 1 lemon, 2 Tablespoons mashed raspberries