In a large bowl, combine flour and salt. Add butter, and using your fingers or a pastry blender, combine until mixture resembles coarse pea-sized crumbles.
In a small bowl, mix together water and vinegar. Pour a little at a time into bowl with flour, and incorporate into dough just until it holds together. Alternatively, you can do this all in a food processor or stand mixer. The dough should hold together but still be a little flaky.
Shape dough into a disc and wrap in plastic. Refrigerate for at least an hour.
Preheat oven to 400F.
Toss potatoes, sweet potatoes, carrots, and onion in olive oil. Transfer to baking sheet and arrange in a single layer. Sprinkle generously with salt. Bake for 30-35 minutes until vegetables are tender.
Meanwhile, combine sugar and water in a small saucepan over medium-high heat. Cook until mixture is a light golden color, about 5-7 minutes, stirring occasionally. Remove from heat and stir in vinegar and a pinch of salt. Immediately pour caramel into a 9 inch pie pan and tilt to spread evenly across the bottom of the pan. Scatter rosemary leaves on top.
Once vegetables are done roasting, arrange the carrots, sweet potatoes, and potatoes in a single layer in the pie pan, filling in the small holes with carrots. Top with onions and crumbled goat cheese.
Roll out dough to a 12 inch round. Drape dough over pie pan and tuck edges in along the sides. Prick dough all over with a fork. Bake for 20 minutes, then reduce heat to 350F and bake for an additional 15-20 minutes, until crust is golden brown.
Let tart cool for about 5 minutes and then invert it onto a plate.