Preheat oven to 375°F.
Whisk together oil, honey, curry powder, and thyme in a small bowl.
Toss with carrots on a rimmed baking sheet; season generously with salt and pepper.
Roast in preheated oven until just tender, 20 to 25 minutes.
Let cool completely, about 15 minutes.
Roll out puff pastry sheet on a lightly floured work surface into a 12- x 9-inch rectangle.
Cut into 12 small rectangles.
If pastry feels sticky, place on a baking sheet, and refrigerate 10 minutes.
Arrange pastry rectangles on a clean work surface.
Whisk together egg and 1 teaspoon water in a bowl.
Brush one short edge of each rectangle with egg mixture.
Brush remaining surface of rectangles evenly with sour cream (about ¾ teaspoon per rectangle).
Place 1 carrot lengthwise on each rectangle on opposite side of egg mixture.
Roll to enclose carrot in pastry; press edges to seal.
Arrange wrapped carrots, seam sides down, on a baking sheet.
Freeze until firm, about 10 minutes.
Transfer wrapped carrots to a parchment paper–lined baking sheet.
Cut a few shallow vents in top of each pastry.
Brush with remaining egg mixture; sprinkle with additional curry powder, salt, and pepper to taste.
Bake at 375°F until pastry is golden and puffed, about 25 minutes.
Transfer to a wire rack; let cool slightly, about 5 minutes.
Serve with additional sour cream.
