Heat the olive oil in a medium skillet over medium-high heat. Add the prosciutto and saute until crisp and browned in places, 2 to 3 minutes; transfer with a slotted spoon to paper towels. Turn the heat down to medium low and add the garlic slices; saute until translucent and barely turning golden, 30 to 60 seconds. Add the vinegar, a big pinch of salt, and a few red chile flakes. Stir and remove the pan from the heat. Leave the dressing in the pan and reheat just before the beets come out of the oven. Add the warm beets and shallots to the pan and toss to coat. Reserving a little of each for garnish, toss the mint and prosciutto into the pan with the beets, stir again, and taste for salt. Turn the beets out onto a small platter and top with the remaining mint and prosciutto.

Heat the oven to 400 degrees F. Measure two 2-foot lengths of aluminum foil and crisscross them on a small rimmed baking sheet. Slice the tops and bottoms off the beets. Cut the beets into wedges from top to bottom (small beets may only need to be quartered; cut larger beets in half horizontally first, then cut each half into about 6 wedges). Each wedge should be about 1-1/2 inches at its widest point. Toss the beets well with the olive oil and salt and put them, in one layer, on the cross section of foil on the baking sheet. Group them snugly but in one layer. Lay the herb sprigs on top of the beets and close the package tightly by folding each sheet of foil in and sealing it with the others. The package should be flat and relatively airtight.

Roast the beets for 1-1/2 hours. Remove them from the oven and, avoiding the steam, carefully peel back the foil. Check to see that the beets are cooked through (a thin knife should slide through without resistance). Also check that the bottoms of the beets are nicely caramelized (they should be darkened and wrinkled). If they're not done, seal the package again and return the beets to the oven for 10 to 15 minutes. Depending on the heat of your oven, the thickness of your baking sheet, and the size of the wedges, they may take more or less time.

Make Ahead:The beets can be made several hours in advance and reheated or served at room temperature. To reheat, spread the wedges on a baking sheet and heat in a 350 degrees F oven for about 10 minutes.

Variations: Slow-Roasted Beets & Shallots: Add 5 or 6 large peeled shallots, cut in half or quartered into wedges through the stem (keep the stem intact), to the beets before roasting, and toss with the oil. Roasting time will be slightly longer because of the moisture the shallots give off.