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1
Preheat the oven to 350F.
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2
Place the meat, carrots, celery, garlic, onion, bay leaves, and peppercorns in a high-sided ovenproof pan with enough room for the jowls to move around and add water to cover by 1 inch.
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3
Cover the pan with foil or a tight-fitting lid and braise for 3 1/2 to 4 hours, or until a paring knife slides through the meat easily.
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4
Remove the jowls from the liquid, place in a shallow pan, and cover with a plate.
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5
Discard the liquid and vegetables.
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6
Put a couple of cans or a cast-iron pot on top of the plate to press the jowls and refrigerate overnight.
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7
Increase the oven temperature to 375F.
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8
Prepare the chestnuts by cutting an X in the pointed end of each with a sharp knife.
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9
Heat a cast-iron skillet as hot as you can get it and place the chestnuts inside in a single layer.
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10
Brown for about 1 minute on the first side.
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11
Using tongs, flip the chestnuts and brown for 1 to 1 1/2 minutes longer.
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12
The color will darken and become spotted with black.
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13
Remove the pan from the heat and let cool.
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14
When the chestnuts are completely cool, the shells should pop off easily.
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15
Quarter the chestnuts and set aside.
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16
Remove any tough outer leaves from the Brussels sprouts by snapping them off at the base.
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17
Trim the base and quarter each sprout.
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18
Trim the stalks of the turnips and peel.
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19
Quarter lengthwise.
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20
Heat the butter in a saute pan over medium-high heat.
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21
Add the turnips and Brussels sprouts and stir to coat in butter.
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22
Cook until golden brown and charred in places, stirring frequently, 5 to 6 minutes total, adding the chestnuts halfway through the cooking time.
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23
While the vegetables are cooking, season the jowls liberally on both sides with kosher salt and plenty of cracked black pepper.
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24
Heat the olive oil in an ovenproof saute pan over high heat and sear the jowls for 5 minutes with the fatty side down.
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25
Pop the pan in the oven and roast for an additional 5 minutes, or until crispy, golden, and heated through.
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26
When the vegetables are done, season with salt and pepper and toss with the thyme.
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27
Serve with the jowls.
