Preheat the oven to 375°. Line a large, rimmed baking sheet with parchment paper. Arrange the bacon slices on the paper and brush both sides liberally with the chutney, leaving any chunks in the jar. Bake the bacon for about 30 minutes, turning once, until caramelized. Transfer the glazed bacon to a rack and let cool.
Meanwhile, light a grill or preheat a grill pan. Rub the chile with oil and grill until charred all over, about 5 minutes. Transfer to a bowl, cover with plastic wrap and let stand for 10 minutes. Peel the poblano and coarsely chop it.
Brush the cut sides of the buns with oil and grill until toasted, about 30 seconds. Brush the tomato slices with oil and grill just until lightly charred, about 1 minute. Place the grilled tomato slices on the bottom half of the buns.
Form the beef into four 1-inch-thick patties and brush with oil. Season with salt and pepper and grill over high heat, turning once, until nearly medium-rare, about 6 minutes. Top with the goat cheese and chopped poblano, close the grill and cook until the cheese is completely melted and the burgers are medium-rare, about 1 minute. Set the burgers on the buns, top with the glazed bacon and watercress sprigs and serve right away.