Step 1
Preheat the oven to 375°F. Turn on the exhaust vent, if you have one.

Step 2
In a large skillet, heat a thin coating of vegetable oil until very hot. In a large bowl, toss the carrots with the salt and flour. Add the carrots to the skillet, and cook over high heat, turning occasionally, until they are deeply browned all over and even black in spots, about 15 minutes. Transfer to a medium ovenproof braising pan, about 10 inches in diameter.

Step 3
Reduce the heat to medium high and add more oil to the skillet, if needed. Add the pearl onions and garlic, and cook until deeply browned and black in spots, 2 to 3 minutes. Transfer to the braising pan.

Step 4
With the heat still on medium high, add the potatoes to the skillet and cook until charred, about 12 minutes. Transfer to the braising pan. Add the leek to the skillet and also cook until charred, about 3½ minutes. Transfer the leek also to the braising pan.

Step 5
Add the wine to the skillet and cook, scraping up any browned and black bits from the bottom. Add the thyme and peppercorns, bring to a boil, and cook over medium-high heat until reduced by half, about 3 minutes.

Step 6
Transfer the reduced wine to the braising pan. Add the demi-glace to the braising pan, then transfer the pan to the oven (uncovered) and cook until the carrots are fork-tender, about 45 minutes. Let cool slightly. Remove and discard the thyme and peppercorns.

Step 7
While the oven is still hot, heat four serving plates. Use a slotted spoon to divide the vegetables among the plates. Set the pan over medium-high heat on the stovetop. When the liquid is just barely simmering, add the butter, slowly, whisking until melted. Spoon the liquid over the vegetables and serve.