Preheat oven to 400°F and place a rack in the middle. Line a rimmed baking sheet with parchment paper or a silicone mat.
Generously flour a work surface and place the chilled dough disk on the flour. Dust the top of the dough with flour. Using a rolling pin, roll dough into a circle, turning and flipping the dough as needed with a bench scraper to prevent sticking, until the circle measures 14 inches wide and 1/8 inch thick. Starting at one edge, roll the dough around the rolling pin and transfer to baking sheet. Chill for 30 minutes.
In a small bowl, combine Skellig, goat cheese, mayonnaise, chives, and thyme. Taste and season with pepper. Spread the mixture in the center of the chilled dough round, leaving a 1 1/2 inch border. Layer the tomato slices in concentric circles, overlapping slightly, over the cheese mixture. Fold the border up and around the tomatoes, pleating it to make a pretty, circular enclosure leaving the center open. Drizzle a little olive oil over tomatoes and season with salt and pepper. Brush the top of the pleated dough with beaten egg.
Place galette in the preheated oven and bake until crust is deep golden brown, about 40 to 45 minutes. Use a metal spatula to lift the galette slightly and check the underside of the crust, which should be a beautiful brown color. Transfer to a rack to cool for 15 minutes. To serve, transfer the galette to a serving plate and garnish with basil.