Adjust oven rack to lower-middle position, set large heatproof serving bowl on rack, and heat oven to 200 degrees. Bring 4 quarts water to rolling boil in large Dutch oven or stockpot.
While water is heating, heat oil in large skillet over medium heat until shimmering, but not smoking. Add bacon and cook, stirring occasionally, until lightly browned and crisp, about 8 minutes. Add wine and simmer until alcohol aroma has cooked off and wine is slightly reduced, 6 to 8 minutes. Remove from heat and cover to keep warm. Beat eggs, cheeses, and garlic together with fork in small bowl; set aside.
When water comes to boil, add pasta and 1 tablespoon table salt; stir to separate pasta. Cook until al dente; reserve 1/3 cup pasta cooking water and drain pasta for about 5 seconds, leaving pasta slightly wet. Transfer drained pasta to warm serving bowl; if pasta is dry, add some reserved cooking water (see below) and toss to moisten. Immediately pour egg mixture over hot pasta, sprinkle with 1 teaspoon sea salt flakes or 3/4 teaspoon table salt; toss well to combine. Pour bacon mixture over pasta, season generously with black pepper, and toss well to combine. Serve immediately.