How to Make It
Preheat oven to 400°F. Cut pears lengthwise into sixths. Toss together pears, shallots, olive oil, 1 teaspoon salt, and 3/4 teaspoon pepper on a rimmed baking sheet. Spread in a single layer, with shallots cut sides up. Bake in preheated oven until shallots and pears are just tender, about 25 minutes. Set aside.
Combine sugar and 2 tablespoons vinegar in a 2-quart (11-inch) ceramic tarte Tatin pan over low, and cook, without stirring, swirling pan occasionally, until mixture is light golden brown, 12 to 15 minutes. Increase heat to medium-low, and swirl in butter, 1 teaspoon salt, and remaining 1/4 cup vinegar. Continue cooking caramel, stirring often with a wooden spoon, until very thick, foamy, and amber colored, 7 to 9 minutes. Remove from heat.
Immediately arrange shallots, cut sides down, in a single layer across caramel in bottom of pan. Nestle pear slices, cut sides down, in any empty spaces. (It will be a tight fit; bottom of pan should be packed.) Sprinkle with lemon zest. Prick pie dough in several places with a fork. Place dough over pear-shallot mixture, tucking edges into pan so it fits snugly around mixture. Bake in preheated oven until golden brown, 28 to 32 minutes. Let cool in pan 20 minutes.
Beat goat cheese, remaining 1/4 teaspoon salt, and remaining 1/4 teaspoon pepper in bowl of a stand mixer fitted with a paddle attachment on medium-high speed until smooth and creamy, about 3 minutes. With mixer running on medium-high speed, gradually stream in heavy cream, beating until mixture is light and fluffy, about 1 minute, stopping to scrape sides of bowl as needed.
If needed, carefully loosen edges of tart with a small knife. Invert tart onto a serving platter. Serve immediately with whipped goat cheese.