Preheat the oven to 400 degrees F. Roll out the puff pastry on a lightly floured surface into a 10-by-12-inch rectangle. Transfer to a baking sheet; use a paring knife to score a 1/2-inch border around the edges. Poke all over with a fork inside the border. Transfer to the oven and bake until deep golden brown, about 20 minutes. Transfer to a rack; gently press down the middle if puffed. Let the crust cool completely.
Meanwhile, working with one type of vegetable at a time, drizzle with olive oil, season with kosher salt and pepper and gently toss in a large bowl. Spread out on parchment paper-lined baking sheets, leaving space between the vegetables so they don't steam. Roast until just tender, about 15 minutes.
Pulse the goat cheese, pea shoots and lemon zest and juice in a food processor until pale green and mostly smooth. Transfer to a bowl; stir in the creme fraeche, 1/4 teaspoon kosher salt and a few grinds of pepper.
Spread the goat cheese mixture on the crust, inside the border. Arrange the vegetables on the tart (they can be warm or room temperature). Sprinkle with coarse sea salt.