Heat 1 1/2-tablespoons unsalted butter and 1 1/2-tablespoons olive oil in a large, heavy saucepan over medium heat. When hot, add 1 1/2-cup chopped shallots (about 1/2 pound or 5 large shallots). Cook, stirring often, until softened and translucent, for 3 to 4 minutes. Add 3/4-pound Yukon Gold or Russet potatoes, peeled and cut into 1/2-inch chunks, and, 1-tablespoon chopped garlic. Cook and stir for another minute.
Add 2 1/4-cups each of chicken stock and whole milk. Bring mixture to a simmer. Reduce heat and cook at a simmer until the potatoes and shallots are tender–about 15 minutes. Gradually whisk in 3/4-cup grated Parmesan cheese. Stir until it has melted. Remove pan from heat, then whisk in a generous pinch (about 1/16-teaspoon) cayenne pepper. Season to taste with Kosher salt–more or less depending on whether or not the chicken stock was low-sodium. If the soup is too thick, stir in 1/4-cup each of additional stock and whole milk. (The soup can be prepared 1 day ahead; cool, cover and refrigerate. reheat over medium heat, while stirring to keep from sticking.)
To serve, ladle into soup bowls, then garnish with some of each of the following: additional Parmesan cheese, 1 1/2-ounces thinly-sliced prosciutto, cut into strips about 3 inches long by 1/4 inches wide that have been sauteed until crisp in 1-tablespoon olive oil, then drained on paper towels, and, 3/4-cup coarse fresh bread crumbs that have been toasted by tossing constantly in 1-tablespoon hot olive oil for 3-4 minutes. (The prosciutto and breadcrumbs can be prepared 4 hours ahead; leave uncovered at a cool room temperature until ready to use.)
Serve with a green salad and fresh crusty bread. C’est magnifique!