Melt butter in a large frying pan over medium-high heat. Add onions and 1/2 tsp. salt. Cook, stirring occasionally, until onions are soft, about 5 minutes. Add apples and stir to coat. Reduce heat to medium and cook, stirring often, until onions are medium brown, about 30 minutes. Stir in pepper and salt to taste. Set aside and let cool, about 10 minutes.
Preheat oven to 375°. Unwrap puff pastry sheet. Cut sheet into 16 squares, then halve those squares diagonally to make 32 triangles. Arrange them, not touching, on a baking sheet.
Place a spoonful of cooled onion-apple mixture on each triangle. Top with a sprinkle of shredded cheese and minced thyme. Bake until puffed and golden, 15 to 20 minutes. Serve hot.
Note: Nutritional analysis is per triangle.