Unroll the thawed puff pastry onto a parchment-lined baking sheet. Using a knife, score a border about half an inch from the edge, being careful not to cut all the way through. This border puffs up around the filling and gives you that classic bakery-style edge.
Spread the Dijon mustard evenly inside the scored border. I find this step is what really elevates the flavor, since it adds a savory depth beneath the cheese. Don't skip it even if you're not usually a mustard fan — it mellows out completely once baked.
Arrange the brie slices across the pastry base, followed by the tomato slices slightly overlapping each other. This layering helps the cheese melt into every gap and keeps the tomatoes from sliding around once baked.
Drizzle the tomatoes with olive oil, then season generously with salt, pepper, and thyme. Brush the exposed pastry border with the beaten egg for a beautiful golden shine once it comes out of the oven.
Bake at 400°F for 22-25 minutes, or until the pastry is puffed and deeply golden. Keep an eye on it during the last five minutes since ovens vary and puff pastry can go from golden to overdone quickly.
Let the tart rest for about 5 minutes before slicing. This helps the cheese set slightly so it doesn't ooze everywhere. Top with fresh basil right before serving for a burst of color and flavor.