Special equipment: Blender
Bring a large pot of water to a boil.
Heat a medium skillet over medium heat. Add the butter and olive oil and let the butter melt. Add half of the basil and let cook until wilted, 1 minute; add the cream and season with salt and pepper. Turn the heat to medium low and simmer gently to reduce and thicken, about 5 minutes.
Generously salt the boiling water. Add the pasta and stir. If using fresh trofie, cook until al dente, 3 to 4 minutes. If using dried trofie, cook according to the package instructions until al dente, usually 8 to 10 minutes.
Pour the basil and cream mixture into a blender and add the remaining basil. Cover, making sure to vent the top and cover with a kitchen towel. Puree until smooth, about 30 seconds. Taste and adjust the seasoning if necessary.
Drain the pasta and return it to the pot. Add half of the basil cream sauce and mix. Transfer to a serving bowl and pour over the remaining sauce. Garnish with the basil sprig and serve immediately.
When blending hot liquid, first let it cool for five minutes or so, then transfer it to a blender, filling only halfway. Put the lid on, leaving one corner open. Cover the lid with a kitchen towel to catch splatters, and pulse until smooth.