Add the yeast, water, and a pinch of the sugar to the bowl of a stand mixer fitted with the hook attachment and let stand until foamy, 8 to 10 minutes.

Whisk the 2 tablespoons of sugar, the flour, and salt in a separate medium bowl and set aside.

Whisk the oil, 2 tablespoons of honey, and 1 egg in a measuring cup and add to the yeast mixture, along with the flour mixture. Mix on low speed until an elastic, uniform dough forms, scraping down the sides of the bowl if needed, 2 to 4 minutes. Turn the dough out onto a clean, dry work surface (the dough should be tacky but not sticky; dust lightly with flour if needed). Knead by hand until the dough is a smooth, satiny ball, 2 to 4 minutes more.

Transfer the dough to a large greased bowl, cover with plastic wrap, and let rise in a warm, draft-free place until doubled in size, about 1 hour.

Meanwhile, peel and core 1 apple and cut it into ¼-inch cubes (you should have 1½ cups). Transfer to a bowl, cover with a damp paper towel, and set aside. Line two baking sheets with parchment paper, and set aside.

Gently punch down the dough and turn it out onto a lightly floured surface. Divide the dough into 3 equal pieces.

Working with one piece of dough at a time, and keeping the other pieces covered with a damp kitchen towel to prevent them from drying out, pat the dough into a 12-inch by 3-inch rectangle with one long side facing you. Sprinkle 1/2 cup of the chopped apple down the center of the dough, leaving about 1 inch clear on each end. Fold the long side closest to you over to meet the other side. Press the edges over the apple to seal in the filling and gently roll the dough into a rope about 24 inches long. Repeat with the remaining dough and apples to create 3 filled ropes that are all the same length.

Transfer 3 of the ropes to each of the prepared baking sheets, aligning them side-by-side. Starting in the middle, braid the ropes loosely—like you’re braiding hair—then pinch both ends of the braid together to create a sealed circle and flip the dough over to hide the seam underneath.

Cover the loaf loosely with plastic wrap or a damp kitchen towel and let sit in a warm, draft-free place until nearly doubled in size, about 30 minutes.

Preheat the oven to 425°F. Peel the remaining apple and slice crosswise with a knife to create thin rounds, about 1/8” thick, discarding the seeds. Transfer to a medium heat-proof bowl and toss with the lemon juice. Set aside.

Combine the remaining ⅓ cup of sugar with ⅓ cup of water in a small saucepan. Bring to a boil over medium-high heat, whisking until the sugar has dissolved completely, about 5 minutes. Pour the syrup over the apple slices and let cool 10 to 15 minutes. Place a fine-mesh sieve over a small bowl and strain the apple slices, reserving the syrup. Set both apples slices and syrup aside.

Lightly beat the remaining egg with 1 tablespoon of water and brush it evenly over the top of the challah. Layer the apple slices over the top of the braided segments of the challah. Brush the apple slices with the egg wash.

Bake the challah for 15 minutes until beginning to darken in places. Reduce the oven temperature to 350°F, remove the challah from the oven and brush again with the remaining egg wash. Return the loaf to the oven and bake until the loaf is a deep golden brown all over, 10 to 12 minutes more.

While the challah is baking, combine ½ cup of reserved syrup with the remaining ½ cup of honey in a small saucepan over medium-high heat. Bring to a boil and cook until reduced to ¾ cup, about 5 to 7 minutes.

Brush the hot challah with all of the hot honey syrup and let cool completely. Serve at room temperature.