Combine cranberries with tomato paste, water, sugar and Dijon in a small saucepan and set over medium-low. Cook, partially covered and stirring occasionally, until cranberries pop, 12 to 15 min. Mash until mixture is thick. Set aside.
Preheat barbecue to medium.
Stir yogurt with bread crumbs and salt in a medium bowl. Crumble in turkey. Mix just until combined. Mixture will feel very wet. Shape into 4 burgers, each about 1/2 in. thick.
Oil grill. Barbecue burgers, with lid closed, until burgers are firm when pressed, about 5 min per side.
Serve burgers on buns. Top with ketchup.