Step 1

Heat 1 1/2 teaspoons oil in a large skillet over medium heat. Add onions, brown sugar, 1/2 teaspoon pepper and 1/4 teaspoon salt. Cook, stirring occasionally and scraping up any browned bits, until the onions are light brown and soft, 15 to 20 minutes.

Step 2

Add wine and bring to a simmer, stirring and scraping. Reduce heat to medium-low and cook, stirring occasionally, until the wine is reduced and the onions are golden brown, 20 to 25 minutes more.

Step 3

Preheat oven to 400 degrees F. Line a large rimmed baking sheet with parchment paper.

Step 4

Transfer the onions to a medium bowl and stir in egg, 1 tablespoon thyme, crushed red pepper to taste and the remaining 1/4 teaspoon each salt and pepper.

Step 5

Lay a sheet of phyllo on the prepared pan. Keep the remaining phyllo covered with plastic wrap and a damp kitchen towel. Combine butter and the remaining 2 tablespoons oil in a small bowl. Lightly brush (or dab) the phyllo with the mixture using a pastry brush. Repeat, layering and brushing the remaining phyllo.

Step 6

Spread the onion mixture evenly over the phyllo, leaving a 3/4-inch border around the edges. Carefully roll the edges up to form the outer rim of the tart. Sprinkle with feta and the remaining 1 teaspoon thyme.

Step 7

Bake the tart until the crust is brown and crispy, 25 to 30 minutes. Let cool on a wire rack for 5 minutes. Serve warm, garnished with more thyme, if desired.