Grate apples into a bowl and toss with lemon juice.

Add butter and brown sugar to a pan over medium-high heat. As butter melts, give the mixture a stir and add the apple.

Cook, stirring, until apples are browned and most of the liquid is evaporated, about 5 minutes. Transfer to a plate, let cool, then refrigerate until very cold, about 20 minutes.

Separate puff pastry, without thawing, into 4 rectangles. Top 2 rectangles with chilled apple mixture and spread to the edges.

Top each rectangle evenly with about 2 ounces of grated Cheddar, saving about 2 ounces for the tops. Align remaining rectangles on top and flatten. Freeze for 10 to 15 minutes.

Preheat oven to 425°F (220°C). Line a baking sheet with silicone mat or parchment.

Cut each rectangle into 6 strips, twist each strip and place on baking sheet. Top with remaining Cheddar and sprinkle with salt.