For the topping, stir together sugar and lemon zest in a small bowl and set aside.
For the muffins, adjust an oven rack to the middle position and preheat oven to 375 degrees. Spray a standard muffin tin with nonstick cooking spray. Rinse one cup frozen berries under cold water and spread on a double layer of paper towels to dry well. Bring remaining one cup berries and 1 teaspoon sugar to simmer in a small saucepan over medium heat. Cook, mashing berries with a spoon several times and stirring frequently, until the berries have broken down and the mixture is thickened and reduced to about 1/3 cup, about 6-8 minutes. Transfer to a small bowl and cool to room temperature, about 10 to 15 minutes.
Whisk the flour, baking powder, and salt together in a large bowl. Toss the berries that have dried on the paper towels in with the flour mixture. Whisk remaining 1 1/8 cups sugar and eggs together in a medium bowl until thick, about 45 seconds. Slowly whisk in the butter and oil until combined. Whisk in the buttermilk and vanilla until combined. Using a rubber spatula, fold egg mixture into flour mixture until just moistened. The batter will be lumpy with a few dry spots of flour - don't overmix. This is the key to moist and tender muffins.
Using ice cream scoop or large spoon, divide the batter equally among prepared muffin cups (batter should completely fill the cups). Spoon a teaspoon of cooked berry mixture into the center of each mound of batter. Using a skewer , gently swirl the berry filling into the batter using a figure eight motion. Sprinkle lemon sugar evenly over the muffins.
Bake until the top of the muffins are golden and just firm, 17-18 minutes. Cool muffins in the muffin tin for 5 minutes, then transfer to a wire rack and cool 5 minutes before serving.