PICKLED SHALLOTS: Bring ¾ cup water, vinegar, sugar, and salt to a boil in a small pot over high heat. Add shallots and boil for 1 minute. Pour shallots and brine into a glass jar (or other nonreactive container) and seal with lid or plastic wrap. Rest for at least 2 hours at room temperature before using.
CHEESE SAUCE: Bring a large pot of water to a boil over high heat. Add potatoes and cook until very soft, about 10 minutes. Pass through a mesh strainer, then measure out ½ cup of potato puree (reserve remainder for another use).
Bring wine to a simmer in a medium saucepan over medium-high heat, then add garlic. Cook 2 to 3 minutes. Add potato puree and stir for 30 seconds. Add 5 tablespoons cream and bring to a simmer, stirring as it thickens slightly. Remove from heat and stir in Taleggio and parmesan.
Combine half the cheese mixture and half the remaining cream in a blender and process until completely smooth. With machine running, slowly add remaining cheese mixture and cream, alternating each ingredient. Return cheese mixture to pan, season with salt as needed, and keep warm over low heat until ready to serve.
TO SERVE: Heat olive oil in a sauté pan. Add potatoes and cook until warmed through.
Pour cheese sauce into a serving dish and lay potatoes on top. Garnish with pickled shallots, purslane, and black pepper. Serve immediately.