4 small, boned chicken breast halves with skin (2 1/4 lbs. total)
2 cups canola oil, divided
4 large shallots, thinly sliced crosswise
3 tablespoons lime juice
1 1/2 tablespoons Thai or Vietnamese fish sauce
1 large Persian cucumber or 1/4 English cucumber
1/3 cup cilantro leaves
How to Make It
Zest 2 large or 3 medium oranges and set oranges aside. In a food processor, whirl zest, lemongrass, 2 1/2 tsp. sugar, the salt, and chile flakes until very finely minced, scraping inside of container several times. Set aside 1 tbsp. In a medium bowl, turn chicken in 1 tbsp. oil to coat. Pat evenly all over with remaining lemongrass mixture. Chill, covered, at least 2 hours.
Meanwhile, in a small saucepan over medium heat, cook shallots in remaining oil (about 2 cups), stirring often, until deep golden brown, 30 to 35 minutes. With a slotted spoon, transfer shallots to paper towels to drain and crisp. Set oil aside.
While shallots cook, cut peel (including all white pith on outside of fruit) from unzested oranges, following curve of fruit. Working over a bowl, cut between inner membranes to free segments. Squeeze juice from membranes into bowl. Tip juice into a measuring cup; if needed, squeeze zested oranges to make about 6 tbsp. juice. Return juice to segments; set aside. Preheat oven to 450°.
Heat a 12-in. ovenproof frying pan over medium-high heat. Swirl 1 tbsp. shallot oil in pan and brown chicken, skin side down, 4 to 7 minutes. Remove from heat and turn chicken over. Roast chicken in oven until no longer pink in center (cut to test), 12 to 16 minutes.
Meanwhile, add lime juice, fish sauce, 1 1/2 tbsp. shallot oil, reserved 1 tbsp. lemongrass mixture, and remaining 1 1/2 tsp. sugar to oranges. (Save remaining oil for other uses.) Slice cucumber into 1/4-in. rounds. Cut slices in half, then into small triangles.
Set chicken on plates. With a slotted spoon, lift oranges in bowl from dressing and arrange over chicken. Scatter cucumber, half of shallots, and the cilantro on top. Pour dressing into a pitcher. Serve chicken immediately, with dressing and more shallots on the side.
Make ahead: Through step 2, up to 1 day, shallots stored airtight.