Make the pastry dough and allow to chill while the rest of the ingredients are prepared.
Thinly slice or dice the yellow onion. Add the olive oil and onions to a large skillet and cook over medium heat until they begin to soften, about 5 minutes.
Slice the mushrooms and add to the skillet with the onion. Raise the heat to medium-high. Cook, stirring often, until the mushrooms begin to soften and release water, 2-3 minutes.
Finely mince the garlic cloves and add to the mushrooms and onion. Season the mixture with kosher salt and black pepper.
Strip the thyme leaves from the stem. Finely chop the sage. Add the thyme leaves, sage, butter, and dijon mustard to the mushrooms.Continue cooking, stirring often, about 6-7 minutes more, until the mushrooms are lightly browned. Stir the white wine into the mushroom mixture and continue cooking until the liquid has nearly completely evaporated. Remove from the heat and allow the mixture to come to room temperature. Preheat the oven to 400°F.
Assemble the galette: Prepare a baking sheet by lining it with parchment paper, then sprinkling ½ tablespoon flour onto the parchment paper. Lightly dust a clean counter with flour. Roll out the pastry dough with a floured rolling pin into a circle, about 15 inches in diameter. The dough should be about ⅛ inch thick evenly across the circle. If the dough deeply cracks outward toward the edges, lightly pull, overlap, and pinch the cracks back together. Some small (1-2 inch) cracks toward the edges are fine.
When finished rolling, transfer the dough to the prepared baking sheet. I like to do this by gently folding the dough circle in half, then in half again, taking care not to crack or break the dough. Then lift the folded dough onto the baking sheet and unfold back into the original circle.
Slice or shred the fontina cheese. Layer the cheese onto the dough, leaving a 2-3 inch border of dough around the edges. Layer the mushroom mixture over the cheese in an even layer.
Gently fold the edges of the dough over the filling. Beat the egg, then brush the beaten egg over the dough. Sprinkle the dough with salt and pepper. Sprinkle additional thyme leaves over the mushrooms if desired.
Bake the galette at 400°F for 45-50 minutes, until the crust is golden and flaky.
Allow the galette to cool for 5-10 minutes, then slice and serve with a fork.