Preheat oven to 425°F (218°C). Spray a 12-count muffin pan with nonstick spray or line with cupcake liners. You may need a 2nd pan as this recipe makes up to 14 muffins, though you can always bake in batches using 1 pan. Set aside.

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

Using a handheld or stand mixer fitted with a paddle attachment, beat the butter, granulated sugar, and brown sugar together on medium-high speed until smooth and creamy, about 3 minutes. (Here’s a helpful tutorial if you need guidance on how to cream butter and sugar.) Add the sour cream, eggs, lemon zest, and vanilla extract, and beat on medium speed until combined. The mixture will look curdled; that’s OK.

Add in the dry ingredients and beat on low speed until almost combined. Add the milk and lemon juice and beat on low speed until combined. The batter is thick. Fold in the blackberries. (See Note if your blackberries are extra juicy.)

Spoon the batter into liners, filling them 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 16–18 minutes or until a toothpick inserted in the center comes out clean. The total time these muffins take in the oven is about 21–23 minutes. Allow the muffins to cool for 5 minutes in the muffin pan, then transfer to a wire rack to continue cooling.

Whisk the confectioners’ sugar, lemon juice, and milk together. Whisk in the blackberry jam until smooth. If using fresh blackberries, mash them with a fork and then whisk into the icing. If desired, add 2–3 more Tablespoons of sifted confectioners’ sugar to thicken the icing. Drizzle over the warm or cooled muffins.

Muffins stay fresh covered at room temperature for a few days or in the refrigerator for up to 1 week.