Add butter, olive oil, garlic, and leeks to a large, high-sided skillet set over medium heat. Cook, stirring occasionally, just until the leeks are tender, approximately 4-5 minutes. Do not allow to brown.
Add potatoes and chicken broth. Increase heat to high and bring to a rapid boil. Immediately reduce heat to just below medium and cover skillet loosely with a lid. Simmer for 25-30 minutes or until potatoes are fork-tender.
While the potatoes are simmering, combine the sour cream and heavy cream in a medium-sized bowl. Season with salt and black pepper, to taste, and stir to combine. Set aside.
Remove skillet from heat and cool for several minutes. Working in batches, if necessary, carefully transfer the contents of the skillet to a high-powered blender. Cover and blend on high until completely smooth, approximately 1-2 minutes.
Remove lid and transfer the purée into a large bowl. Repeat process with remaining contents of the skillet.
Temper the sour cream mixture by adding 2 or 3tablespoons of the soup to the bowl. Stir to combine.
Add the tempered sour cream mixture to the skillet. Season with additional salt and black pepper, as desired, and stir to combine.
Allow soup to cool completely, then transfer to an airtight container. Refrigerate until chilled, approximately 3 hours to overnight.
To serve, divide the chilled soup between four bowls and garnish with fresh chives. Enjoy!