Preheat the oven to 500°. Preheat a pizza stone or generously oil a large baking sheet. In a medium saucepan, melt the butter over moderate heat. Stir in the 3 tablespoons of flour until a paste forms. Gradually pour in the milk, whisking until smooth. Bring the béchamel sauce to a simmer over moderately high heat, whisking constantly, until thickened, about 4 minutes. Reduce the heat to low and cook, whisking often, until no floury taste remains, about 10 minutes. Remove from the heat and stir in the goat cheese until melted. Season the sauce with nutmeg, salt and pepper.
Cut the pizza dough into 4 pieces. On a lightly floured work surface, roll out each piece to a 6-inch round. Dust the rounds with flour and let rest until slightly puffed, about 15 minutes.
Meanwhile, in a large skillet, heat the 2 tablespoons of olive oil. Add the smashed garlic clove and cook over moderate heat until golden, about 2 minutes. Add the mushrooms, cover and cook, stirring occasionally, until tender and browned, about 15 minutes. Discard the garlic. Season the mushrooms with salt and pepper and stir in the thyme leaves.
Roll out one of the dough rounds to form a 9-inch round. Transfer the round to the peel and brush the edge of the dough with olive oil. Spread 1/3 cup of the béchamel over the dough and scatter one-fourth of the mushrooms on top.
Transfer the round to the pizza stone and bake for about 5 minutes, until the edges are crisp and the béchamel is bubbling. Transfer the pizza to a work surface, cut into wedges and serve. Repeat with the remaining dough and toppings.