Grease a fluted pan, pie tin, or shallow springform pan. Roll out the puff pastry to the width of the baking pan. Dust the rolling pin with flour and carefully roll the pastry back onto the rolling pin, then unroll it over the pan. Press the dough into place and crimp the edges. Brush the pastry with the softened butter.
Place the bell peppers over an open flame or under the broiler and completely blacken them on all sides. Put the blackened peppers in a plastic bag, seal it, and allow them to steam for 20 minutes. Remove the peppers from the bag and peel away and discard all black skin. Cut the peppers in half and remove all seeds and veins. Cut into 1/2--inch thick strips.
Cut the eggplant crosswise on the bias into 1/2-inch-thick slices. Brush the eggplant lightly with the clarified butter and season with salt and pepper. Roast the eggplant in the oven for 15 minutes, or until done but not mushy.
Place the crumbled goat cheese on top of the pastry. Top the goat cheese with the eggplant and peppers in a spiral fashion, overlapping each. Sprinkle the Parmesan over the eggplant and peppers.
Bake the tart for about 20 minutes, or until it is golden brown and the edges are puffy. Garnish with the fresh basil leaves.
Whisk together the vinegar, salt, and pepper in a bowl. Slowly add the olive oil while whisking. Toss with the greens and any remaining eggplant or peppers and serve on top of the tart.