Gently stir the water with the yeast in a small bowl, then set it aside until bubbles form on top of the yeast mixture, about 5 minutes.
Sift the flour into the mixing bowl of an electric stand mixer fitted with the dough hook attachment. Mix in the salt. Add the yeast mixture and start mixing gently. With the machine on, add the oil and keep mixing until the dough has formed into a shiny ball. Transfer the dough to a work surface and divide the dough into 4 equal pieces. Shape each dough piece into a ball and set the dough balls on a lightly floured baking sheet. Cover with a slightly dampened towel and set aside for 50 minutes, or until doubled in size.
To caramelize the onions: Melt the butter in a large deep frying pan over high heat until it bubbles, then add the onions and a pinch of salt, and cook until the onions are translucent, about 12 minutes. Add the sugar and cook over medium-low heat, stirring often, until the onions are turning brown and look all around caramelized, about 2 hours. Place the onions in a fine-meshed sieve to drain off the excess butter. Cool completely.
Position a baking stone on the bottom shelf of the oven and preheat the oven to 425°F. Mix the crème fraîche and fromage blanc in a small bowl to blend.
Roll out 1 dough ball on a lightly floured surface into an oval about 3 millimeters thin and about 10 1/2-inches long by 6 1/2-inches wide. Place the dough on a floured pizza paddle. (If a pizza paddle and stone are not available, you can place the formed dough on a lightly oiled sheet tray.) Spread 3 tablespoons of the cream mixture over the rolled out dough. Then scatter about 1/4 cup of the caramelized onions on the top of the cream, spreading it with an offset spatula to cover but without mixing it into the cream too much. Transfer the tart to the preheated baking stone in the oven and bake the tart until you see some golden brown marbling on the tart and the crust is golden brown on the bottom, about 12 minutes. Repeat with the remaining dough and toppings.
Grate enough of the bottarga over the tarts until you see a nice and uniform yellow layer. Sprinkle a small pinch of fleur de sel and white pepper over the tarts. Sprinkle about 1 teaspoon of thymes leaves over each tart, and serve immediately while it’s steaming hot.