Add the basil, baby spinach, garlic, grated Parmesan, and lemon juice to a food processor and pulse until the leaves are broken down.
With the processor running, slowly stream in the olive oil until the pesto reaches a smooth, spreadable consistency; season with salt and pepper to taste.
Spoon about 1/4 cup of the pesto into a bowl and mix it with the softened cream cheese until fully combined.
Lay a tortilla flat and spread half of the cream cheese–pesto mixture over it, leaving about a 1-inch border around the edges.
Layer two slices of turkey and two slices of smoked provolone on top of the spread, then roll the tortilla very tightly into a log.
Repeat with the second tortilla and remaining ingredients.
Optional: wrap each rolled tortilla tightly in plastic wrap and chill about 30 minutes to make slicing easier.
Slice each log into pinwheels and serve.