Preheat oven to 425ºF. Spray a jumbo muffin pan with non-stick cooking spray, or line with paper liners.
In a medium saucepan, melt the butter over medium-low heat. As the butter melts, it’ll start to foam, then turn a clear golden color. Continue to stir the butter, scraping up any browned specks from the bottom of the pan as it starts to brown. It should have a nutty aroma.
Once browned and nutty, turn off the heat and pour the browned butter into a shallow bowl, allowing it to cool for at least 5 minutes.
In a large bowl, whisk the flour, baking powder, baking soda, and salt together until combined; set aside.
In another bowl, whisk both sugars and the eggs together until thickened and combined, about 45 seconds. Whisk in the browned butter and the oil. Add the yogurt and vanilla, and continue whisking until combined.
Using a rubber spatula, gently fold the wet ingredients into the dry ingredients until almost combined. Pour in the blackberries, folding gently so you don't crush the berries. Batter should still be lumpy with a few spots of flour.
Spoon a heaping tablespoon of batter into each well of the prepared muffin pan. Sprinkle a teaspoon of the cinnamon-sugar on top. Spoon another heaping tablespoon of batter on top of the cinnamon-sugar, followed by another teaspoon of the cinnamon-sugar. Spoon more batter on top, filling the muffin well almost all the way. (Note: If you're using a standard size muffin pan, the batter and cinnamon-sugar amounts will be less for each muffin well.)
Evenly distribute the streusel topping among each muffin by sprinkling it on top.
Bake at 425ºF for 5 minutes, then reduce the oven temperature to 375ºF and continue to bake for another 22-25 minutes, or until muffin tops are golden and a toothpick inserted in the center comes out clean.** Allow muffins to cool in the pan for 10 minutes before transferring to a wire rack to cool completely.