In a food processor, pulse flour and salt to combine.Cut one stick of butter into small cubes. Add to flour and pulse until pea-size pieces form. Drizzle in the ice water and process just until the dough starts to form a ball. Remove from food processor and knead dough briefly to blend.

Preheat oven to 425 degrees F.On a floured surface, roll out the dough into a circle. Transfer to a 9-inch round tart pan. Press in evenly. Prick the dough all over with a fork; refrigerate for 10 minutes.

Cover the dough with parchment and fill with pie weights or dried beans. Bake for 10 minutes. Remove the parchment and pie weights; reduce the oven to 375 degrees F. Bake until golden. 15-18 minutes. Transfer the crust to a rack to cool completely.

In a large skillet, melt 1 Tablespoon butter over medium-high heat. Add mushrooms and cook until most of their liquid has evaporated and the mushrooms are soft and brown, about 10 minutes.Transfer mushrooms to a bowl.

Melt the remaining tablespoon of butter in the skillet. Add the spinach and cook until softened, about 6-8 minutes.

In a small bowl, whisk the goat cheese, milk, eggs, and a pinch of pepper. Spread the mixture over the crust. Sprinkle spinach evenly on top of the cheese layer. Cover with the mushrooms.

Return the tart to the oven; bake just until the cheese mixture sets, about 15-18 minutes.

Sprinkle the tarts with parsley and thyme.