Cut 1/2-inch strips from each edge of both rectangles. Brush a 1/2-inch border of the rectangles with the beaten egg. Place the cut strips over the egg-washed edges to form two tart shells with raised borders. Cover with plastic wrap; chill at least 30 minutes.
Meanwhile, toss the tomatoes, thyme sprigs, olive oil, salt and 1/8 teaspoon pepper in a medium bowl. Spread tomatoes on a foil-lined jelly roll pan or baking sheet. Bake 25 to 30 minutes, until tomatoes begin to shrivel and brown. Remove from oven; let cool slightly.
Brush the raised borders of the tart shells with the remaining beaten egg. Spread 1 teaspoon of the anchovy paste, if using, over the bottom of each.
Combine the goat cheese, ricotta, chopped thyme, 1 egg and 1/8 teaspoon pepper in a mixing bowl until blended. Spoon half of the mixture down the center of each pastry rectangle, spreading level and almost to the edges. Sprinkle the Asiago cheese over the ricotta mixture; top with the tomatoes, dividing evenly between the tarts.
Bake for 30 minutes, or until crust is golden. Transfer to racks to cool for 10 minutes. Cut each tart into 8 to 10 slices and serve warm.