Heat 1 Tbs. of the oil in a 3-quart saucepan over medium heat. Add the shallots and 2 tsp. of the coriander seeds and cook until the shallot is softened and the coriander is lightly browned, about 2 minutes. Add 4 cups water, the sugar, 3 Tbs. salt, and 1/2 tsp. pepper and bring to a boil. Add the green beans and carrots and cook until just beginning to soften, about 4 minutes. Remove from the heat, add the vinegar, and set aside.Meanwhile, mix the remaining 1 tsp. coriander seeds, the tea, 1 tsp. salt, and 1/2 tsp. pepper. Sprinkle the chicken breasts all over with the tea mixture and pat to adhere. Put the flour in a shallow dish and the buttermilk in another. Dredge the breasts in the flour, the buttermilk, and then the flour again.
Attach a candy thermometer to a 12-inch skillet. Heat the remaining 3 cups oil in the skillet over medium-high heat to 335°F. Fry the chicken on both sides, flipping once, until deep golden-brown and just cooked through, 8 to 12 minutes total. Transfer to a wire rack to drain briefly. Sprinkle with a pinch of salt. Strain the pickled carrots, green beans, and shallots and serve with the fried chicken.
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