Combine the vinegar, lemon juice, dijon, basil, garlic, and salt in the food process. Process until the basil is a fine texture.
Slowly pour the olive oil into the food processor while processing to emulsify the fat. Combine with the rest of the ingredients. Once the oil is completely combined, stir in the parmesan cheese.
Notes
Don't add the olive oil all at once. It won't emulsify and you will have an oily dressing. You want to slowly add the oil so you break down the fat into tiny particles that won't separate from the rest of the dressing.