Place de-stemmed strawberries, granulated sugar, cornstarch, juice and zest from the lemon, and sea salt in a medium saucepan. Cook over medium heat, stirring regularly until the strawberries release juices, the mixture simmers, and thickens.

After about 5 minutes, once the strawberries are soft, remove the pan from heat. Use a blender or potato masher to break up the strawberry pieces. Add the butter and whisk until melted and fully combined. Cover and chill the filling in the fridge until ready to use.

In a small bowl, combine warm water, 1 teaspoon granulated sugar, and active dry yeast. Stir and let the mixture sit for 10 minutes, or until foamy. If it doesn't foam, the yeast may be inactive.

Meanwhile, in a large bowl, mix together the flour, 1/3 cup granulated sugar, and 1/8 teaspoon sea salt. Make a well in the center and add the very soft butter, eggs, vanilla extract, warm milk, and foamy yeast mixture. Whisk the wet ingredients lightly, then use a large spoon to combine everything into a rough dough.

With your hands, bring the dough together until cohesive. It will be sticky and not completely smooth. Cover the bowl with a dish cloth and set it in a warm area to rise until doubled in size, about 40 minutes. While the dough rises, butter a 9x12-inch baking pan.

Once doubled, punch down the dough, then fold and flatten it a few times to smooth it out. Place on a floured surface and roll into a large rectangle, roughly 18 by 24 inches.

Reserve 1/4 cup of strawberry filling for the glaze. Spread the remaining strawberry filling evenly over the dough, leaving a 1-inch border along all edges. Scatter the cup of chopped fresh strawberries over the filling.

Carefully roll the dough up from the long side to create a log. Using a piece of string or unflavored dental floss, mark the log into 12 evenly sized pieces and slice cleanly through each mark. This helps keep the rolls round and intact. Transfer the slices to the prepared baking pan, placing any filling that leaks out on top of the rolls.

Allow the rolls to rise again in a warm spot until doubled, about 15 minutes. Meanwhile, preheat the oven to 355°F (180°C).

Bake the rolls for 25 to 30 minutes, or until golden on top. Prepare the glaze while they bake.

In a large bowl, beat the softened cream cheese and powdered sugar together until smooth and creamy. Add the reserved 1/4 cup strawberry filling and beat again until fully incorporated.

Let the baked rolls cool in the pan for about 10 minutes. Generously spread all the glaze over the warm rolls, letting it seep into every crevice. Serve while still warm for the best gooey texture.