Preheat the oven to 180°C (350°F). In a large bowl, cream the softened butter and brown sugar together until light and smooth.
Add the eggs, salt, cinnamon, sourdough discard, and applesauce. Mix until well combined.
Add the all-purpose flour and whole wheat flour. Mix until the batter is smooth and no lumps remain.
In a small cup, dissolve the baking soda in 2 tablespoons of hot water. Stir the mixture into the muffin batter until well combined.
Gently fold in the grated apples and blackberries, if using, being careful not to squash the berries.
Grease the muffin cups generously with butter, or line them with paper liners. Divide the batter evenly among the muffin cups.
To prepare the streusel topping, combine the sugar, flour, oats, and a pinch of salt in a bowl. Add the chilled butter and rub it in with your fingertips until the mixture is crumbly and resembles coarse breadcrumbs.
Top each muffin with a generous spoonful of streusel topping—about 1 tablespoon per muffin is just right.
Place the muffin tray in the preheated oven and bake for 12–15 minutes, until the muffins are risen and golden brown on top.
Let the muffins cool in the muffin pan for a few minutes, then gently transfer them to a wire rack to finish cooling.