Instructions: In a small, heavy saucepan over medium heat, melt 9 tablespoons butter. Let cook until white foam begins to sink to bottom and turn a rich nut brown, about 7 minutes. Pour butter through a fine mesh sieve into a measuring cup. You should have 1/2 cup (discard any extra). Refrigerate until butter solidifies, about 3 hours.

Meanwhile, in a large skillet over medium-low heat, melt remaining 2 tablespoons butter with olive oil. Add onions, shallots and thyme, and season with salt and pepper. Toss well and cook gently, stirring occasionally, until onions are soft and caramelized, about 1 hour. Add a splash of water to pan during cooking if onions look dry. Remove thyme branches and stir in vinegar and more salt and pepper. Set aside.

In bowl of a food processor, pulse flour and 1/4 teaspoon salt to combine.

Cut chilled brown butter into pieces and add it to flour. Pulse until mixture resembles oatmeal. Sprinkle 2 to 3 tablespoons of cold water over mixture and pulse until it begins to come together. Empty dough onto a work surface and press it into a disk. Wrap in plastic wrap and chill for 30 minutes.

On a lightly floured surface, roll dough out to a 10 inch round. Fit it into a 9 inch tart pan, folding excess dough back into crust to build up sides. Prick dough all over with a fork and wrap in foil. Refrigerate for at least 30 minutes or overnight.

Preheat oven to 375 degrees. Line tart shell with foil and fill with pie weights or rice. Bake for 15 minutes, then remove foil and weights and bake for 10 minutes longer, until crust is golden. Cool on a rack. Lower oven temperature to 325 degrees.

In a bowl, whisk together creme fraiche and egg yolks. Stir in caramelized onions. Spread mixture in tart shell and bake until set, about 25 to 30 minutes.

Serve warm or at room temperature.

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