Adjust oven rack to middle position and heat oven to 400°. Roll pastry into 10" square on lightly floured counter. Using plate, bowl, or skillet lid as template, cut out 10-inch round. Discard trimming. Poke pastry round all over with paring knife, then transfer round to large plate. Refrigerate until needed.

Trim stem ends of shallots and halve from root end to stem end, leaving skin intact. Trim root ends but leave intact. Brush cut sides with oil and sprinkle with ¹/₈ teaspoon salt and ¹/₈ teaspoon pepper. Arrange cut side down on rimmed baking sheet and roast until tender and paring knife meets no resistance when slipped into center, 30–33 minutes.

While shallots roast, bring onions, water, 1 tablespoon butter, baking soda, ¹/₄ teaspoon salt, and ¹/₄ teaspoon pepper to simmer in 10" ovensafe skillet over medium-high heat. Cover and cook until water has evaporated and onions start to sizzle, 10–12 minutes.

Uncover and, using wooden spoon, spread onions into even layer, pressing into sides and bottom of skillet. Cook, without stirring, for 30 seconds. Stir onions, then spread into even layer, pressing into sides and bottom of skillet. Repeat until dark fond develops in pan. When fond develops, add 1–2 tablespoons water, scrape up browned bits, and stir into onions. Repeat process of letting fond develop and deglazing until onions are very soft, well browned, and slightly sticky, 10–12 minutes. Stir in thyme and transfer onions to bowl. (Onions can be refrigerated for up to 3 days or frozen for up to 1 month.)

In now-empty skillet, bring vinegar, sugar, and remaining ¹/₂ teaspoon salt and ¹/₄ teaspoon pepper to simmer over medium-high heat, swirling skillet to dissolve sugar. Simmer vigorously, swirling skillet occasionally, until consistency resembles that of maple syrup, 3–4 minutes. Stir in remaining 1 tablespoon butter. Remove skillet from heat.

Gently remove outer shallot skin. Arrange shallots, cut side down, in circular pattern around edge of skillet, nestling shallots snugly. Tuck remaining shallots into center (it is not necessary to maintain circular pattern in center; it’s okay if 1 or 2 pieces don’t fit). Spread caramelized onions evenly over top of shallots, leaving outer inch of shallots exposed. Place pastry round on top. Bake until pastry is puffed, crisp, and deep golden brown, 35 to 38 minutes, rotating skillet halfway through baking.

Let tart cool for 8 minutes. Run paring knife around edge of crust to loosen, then invert plate over skillet. Using pot holders, swiftly and carefully invert tart onto plate (if shallots shift or stick to skillet, rearrange with spoon). Let cool for at least 15 minutes.

Whipped Boursin Topping Using hand mixer set at low speed, beat Boursin and sour cream until smooth and creamy, about 30 seconds.

Serve tart warm or at room temperature. Cut into wedges, passing Whipped Boursin separately.