Cut russet potatoes into 1/8-inch-thick rounds; set aside.
Melt butter in large nonstick skillet over medium-high heat. Pour all but 2 tablespoons melted butter into large bowl. Add potatoes and minced thyme to this bowl; toss gently to coat. Season potatoes generously with salt and pepper.
Add mushrooms and garlic to the 2 tablespoons butter remaining in skillet.
Sauté until mushrooms are tender, about 8 minutes. Cool to room temperature.
Position rack in bottom third of oven and preheat to 450°F.
Line bottom of two 9-inch-diameter cake pans with 1 1/2-inch-high sides with parchment paper; brush generously with melted butter.
Arrange 1 layer of potato slices in bottom of each pan, overlapping slightly.
Top each with second layer of potato slices.
Spoon on the mushrooms in each pan.
Cover with remaining potatoes, dividing equally. Add any remaining butter.
Bake potatoes until tender and very brown and crisp on top (bottom will also be brown and crisp), about 40 minutes.
Invert galettes onto platters; peel off parchment. Cut galettes into wedges.
Can be made 6 hours ahead. Let stand at room temperature. Rewarm potatoes in 350°F oven approximately 10 minutes.