Combine flour with cheddar, 1/2 tsp salt and 1 1/3 cups butter in a food processor. Whirl until coarse crumbs form. Add sour cream and egg yolk. Whirl just until dough comes together. Transfer to lightly floured surface. Form dough into a ball, then divide into 8 equal portions. Shape each into a disc, then wrap in plastic wrap. Refrigerate at least 1 hour or up to 3 days, or freeze up to 1 month.
Preheat oven to 425F. Line a large baking sheet with parchment. Slice about 1/2 in. from tops of apples. Trim bottoms of apples so they’ll sit flat. Peel, then scoop out cores with an apple corer or small paring knife and discard. Brush apples with lemon juice. Stir 1/3 cup brown sugar with pecans and cinnamon. Set aside.
Remove dough from refrigerator. Roll each piece into a 9-in.-wide circle. Set an apple in the centre of each. Fill hollowed core with brown sugar mixture. Brush edges of pastry with beaten egg white. Gather edges of pastry to the top of each apple, forming pleats and scrunching seams together at the top to seal. Repeat with remaining dough, apples and brown sugar mixture. Arrange dumplings on parchment, keeping them about 1 in. apart. Brush with more egg white, then sprinkle with remaining 2 tbsp brown sugar. Spray the underside of a piece of foil. Lay over dumplings.
Bake in centre of oven until pastry is golden, about 35 min. Remove foil during the last 5 min of baking.
Melt 1/3 cup butter in a small saucepan over medium. Add 3/4 cup brown sugar. Bring to a gentle boil, whisking occasionally. Cook for 3 min. Stir in vanilla and 1/2 tsp salt until dissolved. Remove from heat and let stand 1 min. Whisk in cream until sauce is smooth. Transfer dumplings into serving dishes and drizzle with caramel sauce.