Begin by gathering and prepping all of your ingredients according to the ingredient list above to have ready and organized for use.
Preheat oven to 400°.
Cut the tops off of the three heads of garlic, drizzle with a touch of olive oil, plus a sprinkle of salt and black pepper, and wrap each head tightly in foil. Roast for 45 minutes, and once cool enough to handle, squeeze the roasted cloves from their papers and set aside.
Place a large pot over medium-high heat, and add in the butter and about 2 to 3 tablespoons of the olive oil. Once melted together, add in the onion, plus a pinch of salt and black pepper, the white pepper and the Herbes de Provence, and saute for about 3 to 4 minutes until translucent.
Next add in the two pressed garlic cloves, plus the reserved roasted garlic, and stir those in.
Add in the cubed potatoes and the chicken stock and stir to combine, and bring to the boil. Cover and simmer the soup for about 25 to 30 minutes, or until the potatoes are fork tender.
Turn off the heat, and use an immersion blender (or use a regular blender, but take care to work in batches as soup is very hot) to puree the soup completely, until smooth.
Then, add in the half and half plus the sour cream, and use the immersion blender once more to incorporate those. (The soup should be velvety and creamy.)
Check to see if any additional salt/pepper is needed, and serve the soup topped with some minced chives, and perhaps some buttery croutons.