Preheat oven to 425°F and grease muffin pan or line with liners.
In a bowl, whisk together flour, baking soda, baking powder, and salt.
In another bowl, beat softened butter on high for 1 minute, then add sugar and mix well.
Add eggs, vanilla, and yogurt; beat on medium for 1 minute, then high until fully combined.
Gradually mix dry ingredients into the wet mixture on low speed.
Add lemon juice, lemon zest, and milk; mix until combined.
Fold in poppy seeds and blackberries gently to avoid breaking berries.
Fill muffin liners about ¾ full with batter.
Bake at 425°F for 5 minutes, then reduce oven to 350°F and bake for 15-18 minutes more.
While muffins bake, combine blackberries, water, sugar, and lemon juice in a pan; simmer until thickened.
Muddle berries, cool, then strain through sieve to remove seeds.
Gradually whisk berry reduction into confectioner's sugar; add remaining lemon juice until glaze is thick and smooth.
Cool muffins 10 minutes in pan, then completely on rack before drizzling glaze.