Combine flour, Kosher salt and butter in the bowl of a food processor fitted with a blade attachment. Pulse until butter resembles a coarse meal.
With the machine running, add 2 tablespoons water and process until just combined. If mixture seems dry, drizzle in more water, a tablespoon at a time, and pulse to combine. Don’t overmix.
Turn out dough onto a floured board, gather into a flat disk and wrap in plastic wrap. Chill 30 minutes.
Roll out dough on a lightly floured surface to a circle between ⅛-inch and ¼-inch thick. Carefully fit it into a 9-inch tart pan with removable sides and gently lift and press the dough into the corners to fit it flush with the bottom surface. Trim off excess dough. Chill 30 minutes.
Preheat oven to 350°F.
Dock tart bottom all over with a fork. Line the shell with foil and fill it with rice or beans. Bake 25 minutes. Remove the beans and foil and bake another 10-15 minutes, until cooked and light golden brown.
Meanwhile, melt 2 tablespoons butter in a large skillet over medium-low heat. Add onion, season with salt and pepper, and toss to coat. Cover and cook, stirring occasionally, until onions are soft, about 10 minutes. Uncover and cook another 10 minutes, stirring often, until light golden brown. Transfer to a bowl and let cool.
Melt remaining 1 tablespoon butter in the same skillet. Add green onions, season with salt and pepper, and toss to coat in butter. Remove from heat.
Combine goat cheese, heavy cream, eggs and ¼ teaspoon Kosher salt in the bowl of a food processor fitted with a blade attachment. Process until smooth and combined, about 20 seconds.
To assemble tart, spread caramelized onions over the bottom of the par-baked tart shell. Pour goat cheese filling overtop. Scatter surface with green onions.
Bake for 30-35 minutes, until filling is firm when shaken. Turn on broiler and broil, watching closely, until green onions are lightly charred, about 1 minute. Serve warm or at room temperature.