Prepare the olives: Drain the green olives thoroughly. Allow the cream cheese to soften before starting.
Combine the ingredients: Add the drained olives, softened cream cheese, mayonnaise, shredded mozzarella, and sour cream to a food processor.
Add the seasonings: Add the Italian seasoning, onion powder, garlic powder, black pepper, and smoked paprika. Reserve the salt until after blending.
Blend: Secure the lid and process until the ingredients are creamy and well combined. Stop to scrape down the sides as needed.
Check the texture: Continue blending briefly if needed, leaving a little olive texture rather than making the mixture completely smooth.
Taste: Taste the dip and add kosher salt only if necessary.
Serve: Transfer to a serving bowl. Garnish with olives, green onions, or parsley if desired. Serve with crackers, bread, or vegetables. Keep refrigerated until serving.