Add 1 cup blueberries and 1 teaspoon sugar to a small saucepan at medium heat. Allow to simmer and cook, mash the berries with the back of a spoon until all of the berries are broken down. Stir frequently and cook until mixture is thickened and reduced to about ¼ cup, about 6 minutes. Set aside and allow to cool to room temperature, about 15 to 20 minutes.

Add flour, baking powder, and salt to a large bowl and whisk.

To another bowl, add 1 1/8 cups sugar and eggs and whisk until thick, about 45 seconds. Whisk in butter and oil slowly, until combined. Then, whisk buttermilk and vanilla in, until combined.

Using rubber spatula or a wooden spoon, fold in egg mixture and remaining cup blueberries into flour mixture until just moistened. (Do not overmix. The batter will be lumpy and that's fine.)

Using an ice cream scoop or cookie scoop, add batter evenly to each of the muffin cups. The muffin cups will be full. Take one teaspoon of the cooled blueberry jam and add to the top of each mound of batter. Using a chopstick or skewer, swirl the jam into the batter a few times. (Cook's Illustrated says to use figure eight motions.)

Add the lemon-sugar topping evenly over all of the muffins.

Bake the muffins until they are just about gold on top, about 17-19 minutes. Halfway through the baking time, be sure to rotate the muffin tin. Allow muffins to cool in the muffin tin for about 2-3 minutes. Using a butter knife or offset spatula, remove the muffins from the tin and allow them to rest at an angle. Cool for another 5 minutes and place on a cooling rack. Serve with softened butter.

These muffins can be stored in a zippered plastic bag for 2-3 days.