Cook noodles in a 6- to 8-quart pot of boiling salted water until just tender (softer than al dente). Drain well in a colander.
While noodles cook, melt butter in a small heavy skillet over low heat and continue to cook until it begins to turn golden brown, about 6 minutes. In the last 30 seconds, add chopped garlic. Remove from heat.
Spread one third of noodles on a large platter and sprinkle with one third of cheese and one third of spinach. Repeat layering with remaining noodles and cheese in 2 batches, then pour brown butter over noodles and toss with 2 forks to combine. Spread noodles on platter and season with pepper.