DOUGH:
3/4 cup all-purpose flour, plus more for dusting
3/4 cup whole-wheat flour
1/2 teaspoon kosher salt
1 stick unsalted butter, cubed and chilled
1/4 cup sour cream
2 tablespoons ice water
1 tablespoon fresh lemon juice
FILLING:
2 tablespoons extra-virgin olive oil
1 large shallot, thinly sliced
Kosher salt
Pepper
1 cup whole-milk ricotta
1 1/2 teaspoons finely grated lemon zest plus 1 teaspoon fresh lemon juice
1 large garlic clove, finely grated
1 teaspoon thyme leaves, plus more for sprinkling
1 teaspoon minced oregano, plus leaves for sprinkling
1/2 teaspoon minced rosemary, plus leaves for sprinkling
1/2 pound acorn squash—seeded, peeled and shaved into ribbons
1/2 pound celery root, peeled and shaved into ribbons
1 small baking potato, peeled and shaved into ribbons
1 large egg beaten with 1 tablespoon water
1/4 cup freshly grated Parmigiano-Reggiano
2 teaspoons honey, warmed
Make the dough In a food processor, combine both flours with the salt and pulse to mix. Add the butter and pulse until pea-size pieces form. Add the sour cream, ice water and lemon juice and pulse until the dough starts to come together. Transfer to a light floured work surface, gather any crumbs and knead until smooth. Pat into a disk, wrap in plastic and refrigerate until chilled, about 1 hour.
Make the filling Preheat the oven to 400°. Line a baking sheet with parchment paper. In a small skillet, heat 1 tablespoon of the olive oil. Add the shallot, season with salt and cook over moderately low heat, stirring, until softened, about 5 minutes. Let cool.
In a small bowl, mix the ricotta with the lemon zest, garlic, the 1 teaspoon of thyme and the minced oregano and rosemary. Season with salt and pepper. In a large bowl, toss the squash with the celery root, potato and remaining 1 tablespoon of olive oil. Season with salt and pepper.
On a lightly floured work surface, roll out the dough to a 13-inch round. Transfer to the prepared baking sheet. Spread the ricotta on the dough, leaving a 1-inch border. Pile the squash mixture on the ricotta and scatter the shallots on top. Fold 1 1/2 inches of the dough edge over the vegetables. Sprinkle with thyme, oregano and rosemary leaves. Brush the dough edge with the beaten egg.
Bake the galette for 15 minutes, until starting to brown. Sprinkle the Parmigiano over the filling and bake for 15 to 20 minutes, until the vegetables are tender and the crust is golden.
In a small bowl, mix the honey with the lemon juice. Drizzle the lemon honey over the galette. Serve warm or at room temperature.