Preheat the oven to 400 degrees F (200 degrees C).

Place 2 pieces of puff pastry onto a silicone-lined baking sheet. Press the short seam of 1 piece onto the other to form the rectangular base of the tart. Cut seams off the other 2 pieces of puff pastry and press them into the edges of the base to act as the border.

Freeze dough until firm, about 10 minutes.

Score border with a knife, making shallow cuts 1/4-inch apart. Press the bottom of a fork onto the seam in the middle and poke holes all over the base.

Bake tart shell until lightly golden and partially puffed, about 10 minutes. Press down on the base using the bottom of a fork. Avoid pressing border. Let cool for 10 minutes.

Spread mustard over the inside of the tart. Arrange tomato slices into a single layer on top and cut a few slices in half to fill in the gaps. Season with salt and pepper. Generously sprinkle herbes de Provence on top. Add Parmigiano-Reggiano cheese and drizzle most of the olive oil on top.

Bake in the preheated oven until pastry is well browned and crispy, 25 to 30 minutes. Cool to room temperature on the baking sheet or transfer to a wire rack using 2 spatulas. Drizzle remaining olive oil over the tart and sprinkle fresh oregano, parsley, and thyme on top.