Process blackberries in a blender until smooth. Strain berries through a sieve into a bowl, and discard solids.
Process berry puree, shallot, and next 4 ingredients in a blender for 30 seconds. With blender running, pour oil through food chute in a slow, steady stream, and process until smooth.
1 cup frozen blackberries, thawed
1 small shallot, chopped
1 teaspoon white balsamic vinegar
1 teaspoon honey
⅛ teaspoon salt
⅛ teaspoon freshly ground black pepper
¼ cup extra-virgin olive oil