• Place a rack in lower third of oven; preheat to 350°. Roll out disk of dough on a lightly floured surface to a 14” round. Transfer to a 9”-diameter pie dish. Lift up edge and let dough slump down into dish. Trim, leaving about 1” overhang. Fold overhang under and crimp. Freeze 15 minutes.
• Line dough with parchment paper or foil, leaving some overhang. Fill with pie weights or dried beans. Bake until crust is dry around edge, 25–30 minutes. Remove parchment and weights and brush entire crust with egg. Bake until crust is dry and set, 10–15 minutes. Let cool.
• Meanwhile, cook garlic in a medium saucepan of boiling salted water until beginning to soften, about 3 minutes; drain.
• Wipe saucepan dry and heat oil in pan over medium-high. Add garlic and cook, stirring occasionally, until cloves start to turn golden brown, about 2 minutes.
• Add vinegar and 1 cup water and bring to a boil. Reduce heat and simmer until garlic is tender, 10–12 minutes. Add maple syrup, rosemary, and thyme, and season with salt and pepper. Cook, stirring occasionally, until liquid is syrupy and coats garlic, about 5 minutes.

For the Mushrooms

• Melt butter in a skillet; add sliced mushrooms and saute until juices are released and starting to brown.

To Assemble

• Scatter cheese over crust; top with spinach and mushrooms. Whisk crème fraîche, cream, and remaining eggs in a medium bowl; season with salt and pepper. Pour over spinach. Scatter garlic on top with any syrup. Bake until custard is set and golden brown in spots, 35–40 minutes. Let cool on a wire rack.