Preheat oven to 425°F (218°C). Spray a 12-count muffin pan with nonstick spray or line with cupcake liners. Set aside.

In a large bowl, toss the flour, poppy seeds, baking soda, baking powder, and salt together. Set aside.

In a large bowl using a handheld or stand mixer fitted with a paddle attachment, beat the butter, sugar, and brown sugar together on high speed until smooth and creamy, about 2 minutes. Scrape down the sides and bottom of the bowl as needed. Add the eggs, sour cream, vanilla extract, lemon zest, and lemon juice. Beat on medium speed for 1 minute, then turn up to high speed and beat until combined. Scrape down the sides and bottom of the bowl as needed. Mixture may look a little chunky and curded; that’s ok.

With the mixer running on low speed, add the dry ingredients and milk into the wet ingredients and beat until no flour pockets remain. Gently fold into the blackberries; they may bleed a little color and that’s ok.

Spoon the batter evenly into each cup or liner, filling each all the way to the top. Bake the muffins for 5 minutes at 425°F (218°C), then, keeping the muffins in the oven, reduce the oven temperature to 350°F (177°C). Bake for an additional 15–18 minutes or until a toothpick inserted in the center comes out clean. The total time these muffins take in the oven is about 22–23 minutes, give or take. Allow the muffins to cool for 5 minutes in the muffin pan, then transfer to a wire rack to continue cooling.

Whisk all of the icing ingredients together and drizzle over warm or cooled muffins.

Cover and store leftover muffins for up to 3 days at room temperature or up to 1 week in the refrigerator.