Chicken in Riesling

INGREDIENTS

• 1 1/2 pound small (2-inch) red potatoes
• 4 medium leeks (white and pale green parts only), finely chopped (2 cups)
• 1 whole chicken (about 3 1/2 pound), backbone discarded and chicken cut French style into 8 pieces (see cooks' note, below)
• 3 tablespoons unsalted butter, divided
• 4 medium carrots, halved diagonally
• 1/2 cup crème fraîche or heavy cream
• 2 tablespoons finely chopped shallot
• 1 cup dry white wine (preferably Alsatian Riesling)
• Fresh lemon juice to taste
• 2 tablespoons finely chopped flat-leaf parsley
• 1 tablespoon vegetable oil