Place a large skillet over medium-high heat and add the oil.
Tie the tenderloin in 3 or 4 different spots using butcher's twine to keep it tight and together. Season all over with the salt. Add to the hot pan and sear until evenly dark brown on all sides, 8 to 10 minutes total. Remove to a plate and let cool to room. Temperature. Chill in the refrigerator for 30 minutes.
Preheat the oven to 425 degrees F.
Place the puff pastry sheet horizontally on a lightly floured board and roll out to a 14-by-10-inch rectangle. Lay the prosciutto slices vertically on the puff pastry, overlapping slightly and leaving a rim exposed pastry around the edges. Spoon the tomato pesto down the center of the prosciutto and gently spread to cover the center of the pastry sheet. Lay the tenderloin in the center of the puff pastry. Brush the edges of the pastry with egg wash, then roll it up around the meat. Seal the edges and tuck them under the meat. Place the wrapped tenderloin on a nonstick baking sheet and brush all over with egg wash. Sprinkle with the flaky sea salt and poke a few small holes in the top of the pastry.
Bake for 40 to 45 minutes, until an instant-read thermometer inserted in the thickest part of the tenderloin reads 125 degrees F. Remove from the oven and let rest for 10 minutes. Slice into thick slices and serve.
The whole Wellington can be wrapped up to 4 hours in advance and stored in the refrigerator before baking.