In a food processor or bowl, mix flour and 1/3 cup sugar until blended. Cut 1/2 cup butter into 1/2-inch chunks and add to flour mixture. Whirl or rub with your fingers until mixture forms fine crumbs. Mix 1 teaspoon vanilla with 1/4 cup cold water. Add to flour mixture and whirl or stir with a fork just until evenly moistened. If mixture is too dry to hold together when pressed, gently mix in 1 more tablespoon water at a time. With your hands, press dough into a ball. Place between two sheets of plastic wrap (each 15 by 15 in.) and roll into an even 1/4-inch-thick round. Chill until firm, about 30 minutes, or up to 1 day.
Meanwhile, rinse and peel apples; quarter lengthwise and cut out and discard cores. Cut quarters in half lengthwise; you should have 8 cups.
Combine remaining 1 cup sugar, 1/4 cup butter, 1 teaspoon vanilla, and lemon juice in a 10- to 11-inch (across the top) ovenproof frying pan (2 1/2- to 3-qt. capacity); set over high heat and stir often until mixture is blended and bubbly, 1 to 2 minutes. Remove from heat.
Arrange a single layer of apples, rounded side down, in a circle around edge of pan, then fill in center to cover bottom; fit remaining apples evenly over the top. Cook over medium-high heat, gently shaking pan occasionally (do not stir), until sugar mixture turns deep gold (juices should bubble up through slices; if they bubble out of pan, reduce heat slightly), 25 to 35 minutes.
Peel one piece of plastic wrap from pastry and invert pastry over apple mixture in pan; peel off remaining plastic wrap. Trim off excess pastry flush with pan rim. Cut 1-inch slits through pastry in 10 to 12 places to let steam escape. Set pan in an 11- by 17-inch baking pan.
Bake in the middle of a 375° regular or convection oven until crust is golden brown and juices are bubbling, 35 to 45 minutes. Let cool 5 to 10 minutes.
Invert a serving plate over tarte; holding pan and plate together, invert again. Lift off pan and fit any apples that are stuck to pan back onto dessert. Cut tarte into wedges and serve hot or warm with vanilla ice cream.