In a medium pot warm the olive oil over medium heat. Add the shallots and cook until tender and starting to brown, about 4 minutes. Add the potatoes and the stock. Bring the stock to a simmer over high heat. Reduce the heat and continue to simmer, covered, until the potatoes are almost tender, about 20 minutes. Add the arugula and lettuce to the pot and continue simmering, uncovered until the greens are tender, another 2 to 3 minutes. Using a slotted spoon, transfer the potatoes and greens to a blender. Pour in enough of the stock to cover the vegetables. Add the cream, salt, and pepper.
*Blend the ingredients together until smooth. Be careful to blend slowly at first with a kitchen towel held tight to the top of the blender lid. Blending hot ingredients can cause the lid to blow off. Pour the blended soup back into the pot with the remaining stock. Stir to combine. Cover and keep warm.
Pour the soup into 4 serving bowls. Top each soup with 1/2-ounce sliced goat cheese. Serve immediately.
* When blending hot liquids: Remove liquid from the heat and allow to cool for at least 5 minutes. Transfer liquid to a blender or food processor and fill it no more than halfway. If using a blender, release one corner of the lid. This prevents the vacuum effect that creates heat explosions. Place a towel over the top of the machine, pulse a few times then process on high speed until smooth.