Heat a large, oven safe pot like a Dutch oven over medium-high heat. Add oil then generously season beef on all sides with salt and pepper. Once the oil is hot, add the beef then sear until a golden brown crust has formed on the bottoms, 1-2 minutes. Turn the beef then sear on the other side. Flip twice more to sear each remaining side then remove beef to a plate and set aside (meat does not need to be cooked through.)
Turn heat down to a touch above medium then add mushrooms, carrots, and shallots. Season with salt and pepper then saute until the mushrooms release their liquid, the liquid evaporates, and the mushrooms begin to turn golden brown, 7-8 minutes. Add garlic then saute until very fragrant, stirring constantly, 1 minute. Add tomato paste then saute for 1 minute stirring constantly. Add red wine then simmer until reduced by half, 3-4 minutes. Use a wooden spoon to scrape up any brown bits from the bottom of the pot.
Add 6 cups beef stock, thyme, and bay leaves then turn heat up to high to bring mixture to a simmer. Add beef + juices back into the pot then place a lid on top and transfer to the oven. Cook for 2-1/2 - 3 hours or until the meat shreds easily with two forks. Transfer beef to a plate then remove thyme stems and bay leaves from the pot.
Meanwhile, make the noodles. Whisk together eggs, extra egg yolk, salt, and 1 Tablespoon water in a small bowl. Add flour to a large bowl then make a well in the center. Pour egg mixture into the well then, using a fork, whisk around the edges to gradually add flour to the egg mixture until everything is wet. Use your fingers to bring the dough together then pour onto a heavily floured surface and press until the dough comes together into a smooth ball. The dough shouldn’t be sticky, but if it’s crumbly add up to an additional Tablespoon water.
Cut dough ball in half then flour a rolling pin and roll each half until very thin, sprinkling with flour and flipping/rotating the dough sheet often to avoid sticking. Using a pizza cutter, cut dough into 2-1/2" by 1/2” strips (no need to be precise!) then toss with extra flour to avoid sticking and place in a container and refrigerate until ready to use.
Place the cooking pot on the stovetop over high heat. Once simmering, sprinkle in the noodles then stir well to ensure none of them are sticking together. Turn heat down to medium then simmer until noodles are tender, 15-20 minutes, stirring often, adding extra beef stock if necessary, and turning the heat down slightly near the end. While the noodles are cooking, shred the beef then add into the pot as you shred it. Once noodles are done, taste then add extra salt and pepper if necessary, then scoop into bowls and serve. Dish will thicken as it sits.