Use the extra butter to grease a 9x13-inch baking pan and a large bowl.
In the bowl of a stand mixer fitted with a paddle attachment, add 1 cup of the warm milk, 1 cup of the granulated sugar, and the yeast and mix to combine.
Let the yeast mixture stand until it is foamy, about 5 minutes.
With the mixer on low speed, add 1/2 cup plus 2 tablespoons of the butter, the eggs, 1 tablespoon of the lemon zest, and 1 tablespoon of the salt to the yeast mixture and beat until it is combined.
Switch the stand mixer to the dough hook attachment.
Add 5 cups of the flour to the dough mixture and beat until a soft, sticky dough forms, about 5 minutes.
Use the extra flour to lightly dust a work surface.
Add the dough to the prepared work surface and knead it a few times. Then, form it into a ball.
Place the dough ball in the prepared large bowl, cover it with plastic wrap, and let it stand in a warm place until it has nearly doubled in size, about 2 hours.
In a large, nonstick skillet over medium-high heat, add the strawberries, the remaining granulated sugar, the remaining lemon zest, 2 tablespoons of the lemon juice, 1/2 teaspoon of the salt, and the black pepper and cook, constantly stirring and mashing the berries, until the juices start to release, about 5 minutes.
Continue to cook the strawberry mixture, stirring frequently, until it has thickened, about 3-5 minutes.
Transfer the saucepan from the heat and let the strawberry mixture cool completely.
Use the extra flour to lightly dust a clean work surface.
Add the risen dough to the prepared work surface and roll it out to a 10x24-inch rectangle.
Spread the strawberry mixture over the dough rectangle and sprinkle with the chopped pistachios.
Starting at one of the longer edges, tightly roll up the dough lengthwise to form a 24-inch-long log, then lightly press the seam to seal.
Cut the dough log into 12 slices.
Place the slices cut-side down and side-by-side in the prepared baking dish.
Cover the baking dish with plastic wrap and let the buns rise in a warm place until they fill the baking dish, about 2 hours.
Preheat the oven to 400 degrees F.
Discard the plastic wrap from the baking dish and bake the buns until they have risen and are golden, about 20-25 minutes.
Transfer the baking dish to a wire rack and let the buns cool, about 30 minutes.
In a medium bowl, add the confectioners' sugar, the remaining melted butter, the remaining milk, the remaining lemon juice, and the extra salt and whisk until it has thickened.
Drizzle the icing over the buns.
Serve warm or at room temperature.