Stir together yeast and warm water in a small bowl and let stand until foamy, about 5 minutes. (If mixture doesn’t foam, start over with new yeast.)

Put 1 ½ cups flour in a medium bowl, then make a well in center of flour and add yeast mixture to well. Stir together egg, 1 tablespoon oil, and 1 ½ teaspoons salt with a fork. Add egg mixture to yeast mixture and mix with a wooden spoon or your fingertips, gradually incorporating flour, until a soft dough forms. Transfer dough to a floured surface and knead, working in additional flour (up to ¼ cup) as necessary, until smooth and elastic, about 5 minutes. (DT: I used a lot of flour on my board to keep the dough from sticking.)  Transfer dough to an oiled bowl and turn to coat. Cover with plastic wrap and let rise in a draft-free place at warm room temperature until doubled, 1 to 1 ½ hours.

While dough rises, heat remaining 1/3 cup oil in a 12-inch heavy skillet over medium-high heat until it shimmers, then sauté fennel seeds until a shade darker, about 30 seconds. Stir in onions, remaining teaspoon salt, and 1/2 teaspoon pepper, then reduce heat to medium-low and cover onions directly with a round of parchment paper. Cook, stirring occasionally, until onions are very tender and golden brown, 1 to 1¼ hours.

Preheat oven to 375°F with rack in middle.

Knead dough gently on a floured surface with floured hands to deflate.  Transfer dough to a heavy baking sheet lined with parchment paper.   Pat out dough into a roughly 15- by 12-inch rectangle, turning up or crimping edge, then brush mustard evenly over dough, leaving a ½-inch border around edge. Spread onions evenly over mustard, then sprinkle evenly with cheese.

Bake tart until crust is golden brown, 30 to 35 minutes. Cut into 2-inch squares or diamonds and serve warm or at room temperature.