1. In a medium saucepan over medium-high heat, stir together sugar, corn syrup, and water until dissolved, 2 to 3 minutes. Cook mixture without stirring until it begins to brown, 5 to 10 minutes. (Use a pastry brush dipped in water to occasionally wash down sides of pot.) When mixture reaches preferred hue (the darker it is, the more flavorful it will be—we prefer a deep-amber color), turn off heat, and, standing at arm’s length, add cream and butter, stirring to combine. The caramel will bubble vigorously at this point, but keep stirring until it subsides. Once butter and cream are incorporated, add salt and stir. Transfer to a heatproof jar and set aside.
1. Add flour and salt to bowl of a food processor and pulse once to mix. Add butter and pulse until no pieces larger than a pea remain, about 12 to 15 1-second pulses. Add smoked cheddar and pulse twice more to combine. Add half the ice water and pulse 3 times. Add remaining water and repeat. Dough should form large clumps and no dry flour should remain. (Add more ice water, a tablespoon at a time, if needed, until dough holds together.)
2. Turn dough onto a floured surface and divide into 2 equal portions. Flatten each into a 1- inch-thick disk, wrap, and refrigerate at least an hour or overnight. When ready to bake, sprinkle 2 pieces of parchment paper with flour. Roll each ball of dough atop parchment paper into a 12-inch round. Line a deep-dish pie plate with one disk of dough. Leave other disk of dough sandwiched between parchment paper sheets. Refrigerate dough while preparing filling.
1. In a large bowl, mix together sugars, salt, and cinnamon. Toss with apple slices and add lemon juice. Transfer apple mixture to a large, heavy bottomed pot, and cook over medium heat until apples are tender when poked with a fork, about 15 minutes.
2. Pour cooked apples into a colander set over a large bowl. Shake colander to drain apple juices. Reserve ¼ cup apple liquid.
1. Heat oven to 400°F. Remove dough from fridge. Pour apples and reserved juice into doughlined pie plate. Remove parchment from dough disk and place dough on top of apples if forming a solid top crust. Pinch together two layers of dough and crimp edges with fingers or a fork. Cut several slits into top crust of pie to vent steam while the pie bakes.
2. Alternatively, if forming a lattice, lay dough on work surface and remove top piece of parchment. Use a pizza cutter or knife to cut 8 to 10 even strips of pastry. Place 1 strip horizontally across bottom of pie plate (closest to you). Then, starting on the left, place a strip vertically, laying it over the first strip of pastry. Add another horizontal strip, laying it over the vertical strip. Lift up edge
3. of horizontal strip closest to you and add a second vertical strip. Lift edge of first vertical strip and lay down third horizontal strip. Repeat to weave a full lattice.
4. Brush top crust of pie with beaten egg. Bake 30 to 35 minutes until crust is dark golden brown. Remove from oven and cool on a wire rack at least 2 hours.
1. Drizzle each pie slice with warmed caramel sauce and sprinkle with sea salt.