Heat oil in a skillet over medium heat; cook and stir onion, garlic, and pepper in hot oil until onion is golden, 10 to 15 minutes. Stir pumpkin into onion mixture and cook until pumpkin is tender, 15 to 20 minutes.
Stir cream cheese, mayonnaise, and basil puree together in a microwave-safe bowl; microwave until warmed and smooth, about 1 minute. Pour cheese mixture over pumpkin mixture and toss to coat; cook until warmed, about 5 minutes.
