For the dough, in the bowl of a food processor, combine the flour, salt and sugar. Add the butter and process until the mixture resembles a coarse meal, 8 to 10 seconds. With the machine running, add the ice water in a slow and steady stream through the feed tube.
Pulse until the dough holds together without being wet or sticky; be careful not to process more than 30 seconds. To test, squeeze a small amount together: If it is crumbly, add more ice water, 1 tablespoon at a time.
Form into a ball and flatten into a disc. Wrap the dough in plastic, transfer to the refrigerator and chill for at least 1 hour and up to 1 day.
For the puree, in a medium saucepan, bring the parsnips, garlic, cream, milk and butter to a boil over medium-high heat. Reduce the heat, cover and simmer until the parsnips are very soft, 10 to 15 minutes. Uncover and cook until the liquid is reduced by half, about 5 minutes, then season with salt. Puree in a blender until smooth and reserve in a container for assembly.
For the parsnips, blanch the rounds in salted water until just barely tender. Remove and shock in a salted ice bath, then drain and reserve for assembly. To assemble the tart, preheat the oven to 400°F, with a rack on the second lowest shelf.
On a lightly floured surface, roll the dough into a 12-inch round. Drape over an 8- or 9-inch pie tin or a tart mold, tucking the edges into the pan. Prick the dough all over with a fork. Bake (pie weights optional) until the crust looks dry, about 20 minutes, then reduce the heat to 350°F and bake until golden brown, 15 to 20 minutes more. Chill the crust and set the oven to 400°F.
Spread a few spoonfuls of the parsnip puree across the top of the tart shell in a thin, even layer. Shingle the blanched parsnips, starting with the smallest pieces at the center and working your way toward the outer edge with the largest pieces. Brush with a little melted butter and season with salt.
Bake until the top of the tart begins to brown slightly, about 15 minutes. Let the tart cool for about 5 minutes before carefully transferring to a large plate. Serve warm, at room temperature or chilled.
