Preheat the oven to 400 degrees F. Roll out the puff pastry into an 11-inch square on a sheet of parchment paper. Transfer the dough on the parchment to a baking sheet. Beat the egg with a splash of heavy cream. Brush a 1-inch border of the egg wash around the edges of the dough, then fold in each about 1/2 inch to make a thin border and press to seal. Prick the center of the dough all over with a fork, then brush all over with the egg wash.
Bake the dough until crisp and golden brown, about 25 minutes. Prick the center of the dough with a paring knife and gently press with the bottom of a measuring cup to flatten, if necessary. Remove to a rack and let cool.
Meanwhile, soak the sliced radishes in a large bowl of ice water until crisp and just starting to curl, about 15 minutes; drain and pat dry with paper towels.
Mix the garlic-herb cheese spread and heavy cream in a medium bowl until smooth. Whisk the vinegar, tarragon, parsley and a pinch of kosher salt in a large bowl. Add the radishes and smoked salmon and toss to coat.
Spread the cheese mixture on the cooled crust, then pile the radishes and smoked salmon on top. Sprinkle with flaky sea salt.