Special equipment: food mill
Preheat the oven to 250 degrees F.
Pat the chicken breasts dry with a paper towel. Coat a medium saute pan with olive oil, and place over medium-high heat. When the pan is hot, sprinkle the skin of the chicken generously with salt, and then place skin-side down into the pan. Cook until the skin is golden and crispy, 4 to 6 minutes. Sprinkle the opposite side of the breast with salt, and then flip the chicken and continue cooking until cooked through, another 5 to 6 minutes. If the pan starts to smoke, lower the heat. Transfer the breast to a sheet pan, tent loosely with foil and place in the oven until the sauce is finished and you are ready to plate.
Remove the excess oil from the pan before returning over medium-high heat. Add the clementine juice, clementine supremes and vinegar, scraping up the brown bits that cling to the bottom of the pan. Stir in the thyme and chiles and let the liquid reduce by half before pouring in the chicken stock and a pinch salt. Continue cooking until the stock has reduced. Taste the sauce, making sure it's delicious, and then remove the thyme stalk and season with more salt, if needed. The sauce should be thickened and a little chunky.
Lay the Yukon Gold and Cauliflower puree on the bottom of the plate, and then slice the chicken breast and fan the slices on top. Spoon some of the sauce on and around the chicken, and garnish with the supreme clementines.
Halve the potatoes, and then cut each half into 6 to 8 pieces. Place the potatoes in a pot large enough to hold the cauliflower as well, and then cover the potatoes by 2 inches of water. Sprinkle generously with salt (it should taste like the ocean) and bring to a boil. After the water has been boiling about 10 minutes, add the cauliflower and cook until the vegetables are fork-tender.
Bring the cream to a scald in a small saucepan. Meanwhile, pass the potatoes and cauliflower through a food mill into a large bowl. Add about half of the hot cream and 1/2 of the butter, and stir with purpose until thoroughly combined. Repeat this process with the remaining cream and butter until incorporated. Taste for seasoning (you will probably have to add salt), and serve warm. Yield: 2 servings.
A skin-on chicken breast with wing bone attached and wing tip and tenderloin removed is also known as an airline chicken breast.