1. Generously brush a 9-in. pie pan with some butter. Preheat oven to 400° (if using metal pans) or 425° (if using glass or ceramic) with a rack set in lower third of oven.
2. Brown bacon in a medium frying pan until crisp, 6 to 8 minutes. Drain on paper towels, then chop.
3. Combine bacon, cheese, flour, minced thyme, salt, and pepper in a small bowl.
4. Thinly slice turnips into rounds with a handheld slicer. Arrange one-sixth of turnips in a layer in pie pan, starting from center, working outward in concentric circles, and slightly overlapping slices. Evenly sprinkle a heaping 2 tbsp. bacon-cheese mixture over turnips and drizzle with about 2 tsp. butter. Repeat to use all ingredients, ending with turnips.
5. Lightly butter bottom of another 9-in. pie pan and set on top of turnips. Fill upper pan with pie weights or dried beans; set pans on a rimmed baking sheet to catch bubbling butter.
6. Bake until edge turns golden brown, 50 to 55 minutes. Carefully remove top pie pan and weights and continue to bake Anna until browned on top, 10 to 15 minutes more.
7. Loosen Anna from pan with a knife and invert onto a plate. Top with thyme sprigs.