Add blackberries, lemon juice, sugar, and cornstarch to a small saucepan over medium heat.
Bring to a simmer then reduce to low and simmer, stirring often, for 10 minutes. Remove from heat and set aside to cool completely.
Combine lemon zest and sugar in a small bowl. Press zest and sugar together with your hands to release the oils from the zest. Set aside.
In a large bowl or the bowl of a stand mixer, dissolve the yeast in warm milk and let stand 3 to 5 minutes; stir.
Add the egg, lemon extract, lemon zest/sugar mixture, flour, salt, and butter, mixing after each addition.
Beat well until smooth, adding more flour a little at a time until dough pulls away from the sides of the bowl.
Knead dough on a lightly floured surface (or in the stand mixer using the hook attachment) for about 10 minutes, until soft and smooth.
Place in a bowl in a warm spot and cover with plastic wrap to rise for about 40 minutes, or until the dough has doubled in size.
Line a 9x9 baking pan with parchment paper and spray with cooking spray.
Roll dough out on a piece of parchment paper dusted with flour or a nonstick pastry mat, forming a rectangle that is about 8 inches by 14 inches.
Spread butter onto dough, and sprinkle with brown sugar. Gently spread cooled blackberry mixture onto dough.
Roll dough away from you into one long roll, using the parchment paper to assist.
Use a serrated knife to cut into 9 even pieces and carefully place into prepared 9x9 baking pan. Cover with plastic wrap or a towel and allow to rise in a warm place until doubled in size again, approximately another 40 minutes.
Preheat oven to 375°F
Bake 20-25 minutes, or until rolls are a nice golden brown.
Remove rolls from the oven and let cool for 5 minutes.
Beat softened cream cheese and butter in a bowl until smooth and creamy, about 2 minutes. Add lemon zest and juice and mix to combine.
Gradually add in powdered sugar, mixing on low, until combined. Beat another minute to a smooth and creamy consistency.
Spread frosting over warm rolls and serve immediately.