Crust:

Blend first 3 ingredients in processor 5 seconds.

Add chilled butter and blend until coarse meal forms; add creme fraiche.

Blend, using on/off turns, until moist clumps form.

Gather dough into ball; flatten into disk.

Wrap; chill 1 hour.

Roll dough out on lightly floured surface to 11-inch-diameter round.

Slide onto rimless baking sheet, cover, and chill until ready to use, up to 1 day.

Filling:

Boil pomegranate juice in heavy large saucepan until reduced to scant 1 cup syrup, about 15 minutes.

DO AHEAD: Pomegranate syrup can be made 1 day ahead. Cover; chill.

Position rack in center of oven and preheat to 400°F

Spread room-temperature butter evenly over bottom and up sides of heavy medium oven-proof skillet (10 inches across top; 8 inches across bottom; 21/4 inches deep).

Sprinkle sugar evenly over butter.

Cook over medium heat without stirring until mixture bubbles all over, about 3 minutes.

Remove from heat.

Stand apple quarters on 1 end around edge of skillet, leaning 1 cut edge against pan side and fitting snugly.

Stand as many apples in center as will fit.

Sprinkle apples with allspice and salt.

Cook apples over medium-high heat without stirring until thick, deep-amber syrup bubbles up, adding any remaining apples as space permits (apples will shrink as they cook), about 20 minutes.

Pour 1/4 cup pomegranate syrup over (mixture will bubble).

Cook until juices thicken again, 4 to 5 minutes (syrup will be deep amber).

Remove from heat.

Using spatula, press apples gently toward center, then down to compact.

Slide crust over apples.

Press crust down around apples at edge of skillet.

Cut 4 slits in top for steam to escape.

Bake tart until crust is brown and juices at edge are thick and dark scarlet in color, 25 to 30 minutes.

Remove skillet from oven; let stand 1 minute.

Place large plate over skillet.

Using oven mitts and holding plate and skillet firmly together, invert tart onto plate.

Carefully lift off skillet.

Return any apples to tart that may be stuck in skillet.

Let cool at least 15 minutes.

Serve tarte Tatin warm or at room temperature with vanilla ice cream and drizzle of remaining pomegranate syrup.