In a large bowl, gently toss the tomatoes with 1 teaspoon salt. Transfer tomatoes to a colander and set colander in sink. Let tomatoes drain for 30 minutes.
Meanwhile, adjust oven rack to lower-middle position and heat oven to 375°F. Line a rimmed baking sheet with parchment paper. Let chilled dough sit on counter for about 10 minutes to soften slightly. On a lightly floured surface, roll dough into a 12-inch circle, then transfer to the prepared baking sheet (dough may run up lip of sheet slightly; that's ok).
Line another baking sheet with paper towels. Shake colander well rid of excess juice. Transfer tomatoes to paper towels to remove more juice. Using the empty bowl from salting the tomatoes, combine the tomatoes, shallot, oil, thyme, garlic, pepper and remaining ½ teaspoon salt. Spread mustard over dough, leaving 1 ½-inch border. Sprinkle Gruyère in an even layer over the mustard. Shingle the tomatoes mixture on top of the Gruyère in concentric circles, keeping with in the 1 ½-inch border. Sprinkle Parmesan over tomato mixture.
Carefully grasp 1 edge of dough and fold up about 1-inch over the filling. Repeat around circumference of galette, overlapping dough every 2-inches, gently pinching pleated dough to secure. Brush folded dough with egg (you won't need it all).
Bake until the crust is golden brown and the tomatoes are bubbling, about 45-50 minutes. Transfer sheet to wire rack and let the galette cool for 10 minutes. Using large metal spatula (I use a cake spatula) loosen galette from the parchment and carefully slide onto a wire rack; let cool until just warm, about 20 minutes. Sprinkle with basil. Cut into wedges and serve. Enjoy!