Preheat the oven to 350°F. Line a 12 cup muffin tin with paper cupcake liners and set aside.
Add the flour, granulated sugar, cocoa powder, baking soda and salt to a large mixing bowl. Whisk until well combined and set aside.
Add the milk, oil, eggs and vanilla extract to another large mixing bowl. Whisk until well combined.
Pour the wet ingredients into the mixing bowl with the dry ingredients. Mix until just combined. Add the hot water, and gently mix until well combined.
Divide the cake batter between the paper cupcake liners. Bake for 20 minutes, or until cooked though. Allow to cool in the tin for 5 minutes, before transferring to a cooling rack to cool completely.
Add the blackberries and lemon juice to a small saucepan over a low-medium heat. Bring to a gentle simmer, and continue simmering for 5 minutes or until the blackberries have softened. Mash gently with a fork to release their juices. Strain the blackberries, keeping the liquid and discarding the solids.
Add the softened butter, vanilla extract and salt to the bowl of a stand mixer. Beat on a high speed using a wire whisk attachment for 2 minutes, or until pale and creamy. Add the powdered sugar in 2 batches, and beat until well incorporated.
Add ¼ cup of the blackberry liquid to the frosting mixture, and beat on a medium speed, increasing to a high speed, until well combined and the buttercream is fluffy.
Transfer the buttercream to a piping bag fitted with your preferred piping tip, and pipe onto the cooled cupcakes.
Garnish each cupcake by pressing a blackberry and small sprig of thyme leaves into the buttercream before serving.