Preheat the oven to 350°. In a medium saucepan, bring the shallots and oil to a boil. Cook over moderately low heat, stirring, until the shallots are deep golden, about 18 minutes. Strain the shallots, then drain on paper towels and let cool. Reserve the shallot oil.
Meanwhile, brush the rough side of the pitas with some of the shallot oil and arrange on baking sheets, oiled side up. Reserve the remaining oil for another use. Sprinkle the pitas with salt and cayenne and bake for 10 minutes, or until golden and crisp. Let cool, then break into large pieces.
In a food processor, puree the goat cheese with the sour cream and milk. Season with salt and cayenne; process until smooth. Transfer to a bowl. Stir in three-quarters of the shallots and the parsley. Top with the remaining shallots and serve with the pita crisps.
Make Ahead: The crispy shallots and pitas can be stored in airtight containers for 3 days; the goat cheese mixture can be refrigerated for 3 days.