Heat a few drops of olive oil in a saucepan over medium heat. Add spinach and cook, stirring constantly with tongs, until leaves start to wilt and turn dark green, 2 to 3 minutes. Transfer to a strainer set over a bowl; let cool slightly, about 5 minutes. Wrap spinach up in a clean, dry towel and squeeze out as much liquid as possible.

Transfer spinach into a blender. Add eggs; pulse on and off until pureed. Press puree through a strainer into a bowl if desired.

Pile flour onto your work surface. Make a large well in the center; add salt, 2 teaspoons oil, and the liquid spinach mixture. Stir some flour into the well until liquid is mostly absorbed. Gather in enough flour to form a shaggy dough. Scrape excess flour to the side. Knead dough for a few minutes until smooth and soft but still moldable. Add extra flour or water if needed.

Wrap dough in plastic wrap and refrigerate for at least 1 hour.

Cut dough into 4 pieces; keep 3 pieces wrapped in plastic. Roll 1 piece into a tube and press down into a rectangle about 1/4-inch thick, dusting with semolina flour.

Run dough twice through a stand mixer fitted with a pasta attachment using the thickest setting. Dust with semolina; fold in the rounded ends to retain the rectangular shape. Run pasta sheet through the thickest setting 2 more times. Decrease thickness by 1 setting and run pasta sheet through 2 times. Repeat 2 more times. Dust with semolina if sticky and clean up any edges as needed.

Cut pasta sheets in half lengthwise and sprinkle with semolina. Let dry for 15 minutes. Roll and dry remaining pieces of dough. Run each sheet through the fettuccine setting and dust with more semolina.

Bring a large pot of lightly salted water to a boil. Cook fresh pasta until tender yet firm to the bite, 45 to 60 seconds. Drain most of the water and mix pasta with any piping-hot sauce you desire.