Preheat the oven to 425°F and line a 12-cup muffin tin with paper liners. I like to take some coconut oil on my hands and rub it over the top surface of the tin to ensure that if the muffins grow over the edge, they will not stick.
If using a fresh lemon, zest it and use your fingers to massage the zest into the white sugar until fragrant. This will add a subtle lemon flavour to the muffins.
Add the butter to the bowl of a stand mixer fitted with the paddle attachment. If you don't have a stand mixer, place the butter in a large bowl and use a hand mixer with beaters. Beat the butter on medium-high speed for 3 minutes, until lighter in colour. Add both sugars to the bowl and beat for another 3 minutes, until fluffy and completely dissolved into the butter. Stop to scrape down the sides and bottom of the bowl with a rubber spatula as needed.
Beat in the eggs one at a time, followed by the sour cream and milk. Mix in the lemon juice and vanilla extract. The batter may look curdled but don't worry. See step-by-step photos in the blog post.
Add the flours, cinnamon, baking soda, baking powder, and salt to the same bowl. Gently stir together on top with a spatula, then mix the wet and the dry ingredients until just combined. The batter will be fairly thick. Gently stir in the halved blackberries.
Use a large ice cream scoop (equal to about ¼ cup measuring cup) to fill the lined muffin tin. Distribute the batter evenly among 12 muffins. They should be filled to the top.
Baking Tip: To achieve smooth, round muffin tops, take a spoon and run it under cold water. Dry lightly, then run the back of the spoon over the top of a muffin. This will smooth any spikey bits on the muffin top that could burn. Repeat, rinsing the back of the spoon before each muffin.
Add extra halved blackberries to the muffin tops and sprinkle with coarse sugar if desired. This will result in a nice crunchy muffin top.
Bake at 425ºF for 8 minutes, then without opening the oven door, turn the temperature down to 350ºF and bake for an additional 11-12 minutes until a toothpick inserted in a middle muffin comes out clean. Allow to cool in the pan for 15 minutes before removing to a wire rack. Enjoy warm or allow to cool completely.