Heat a skillet over medium heat; add the olive oil. After the oil heats up, add the shallots. Reduce the heat to low and let the shallots caramelize for about 5 minutes, stirring often. After they’re caramelized, let the shallots cool in the pan.
To make the pastry, add the flour and salt into the bowl of the food processor. Pulse four times. Add the butter, and pulse four more times. Add the cheese and sun-dried tomatoes and pulse four more times. Add 1/4 cup ice water and pulse another four times. If it is very dry, add 1 tablespoon of ice water at a time. The dough will be lumpy.
Turn the contents of the food processor bowl out onto a clean counter and press it together into a ball, working it as little as possible to get it to stick together. Add a little extra water, if necessary. Once you have a cohesive ball, flatten it into a disc, about 5″ in diameter. Wrap the disc in plastic wrap and refrigerate it for a half hour.
When the dough is done chilling, preheat the oven to 400˚F. Line 2 baking sheets with parchment paper. Generously dust a large clean surface with flour then place the chilled dough on the flour. Roll the dough out to a 16-inch square. Cut the square into 4 quarters, forming 8 x 8-inch squares. Note: Save the scraps and bake after the pop tarts for crackers for about 15 minutes.
Place 1/4 of the tomatoes, shallots, and cheddar onto one side of each square, making sure there’s about a 1-inch border on the 3 closest edges. Sprinkle oregano on tomato mixture.
Brush the edges of the dough with the egg wash, and then fold the pastry dough over the tomato mixture. Trim the edges if necessary and seal the edges all the way around with a fork. Move the tarts to the baking sheets and brush the tops with more egg wash. Repeat with the remaining three squares of pastry. Place 2 tarts per baking sheet.
Bake in the oven for 15 minutes, then rotate and turn the baking sheets 180˚. Bake another 10 to 15 minutes or until the pastry is golden brown. Let cool for 10 minutes on the baking pan, then move each tart with a large spatula, to a wire cooling rack. Serve warm or at room temperature alongside a simple green salad.