To make the crackers: Heat the oven to 400 F. Line a baking sheet with parchment paper.
In a food processor, combine the flour, Parmesan, butter, rosemary and salt. Pulse until flour and butter are combined. Add the cream and process until the dough starts to come together and form a ball, about 15 to 25 seconds. If the dough needs more liquid, continue to add a teaspoon of cream at a time, until the mixture holds together but is not sticky.
On a lightly floured surface, roll the dough into a rectangle until it is ½-inch thick. Transfer sheet of dough to parchment paper-lined baking sheet. Continue rolling the dough until it is ⅛-inch thick.
Lightly score the dough with a pizza cutter, creating rectangle crackers. Sprinkle the dough with additional grated Parmesan and flaky salt.
Bake until lightly browned, about 6 to 8 minutes. Cool on a rack, and then break the crackers along the scored lines.
To make the caramelized shallot dip: In a medium skillet, heat oil over medium. Add shallots and season generously with salt and pepper. Cook, stirring occasionally and reducing heat if needed, until shallots are golden brown and tender, 15 to 18 minutes. Let cool slightly.
In a medium bowl, mix shallots, vinegar and yogurt. Season with salt and pepper. Stir chives into the yogurt.
Garnish the dip with additional chopped chives. Serve with the crackers.
Crackers can be stored in an airtight container for up to 5 days. The dip can be stored in an airtight container, refrigerated, for up to 3 days.