Pulse the dry spinach and flour in a food processor until finely combined and a dough begins to gather. Stop and squeeze a handful. If dry and crumbly, add water a teaspoon at a time; if sticky, add flour a teaspoon at a time. Knead briefly until cohesive, cover and rest for 15 minutes.

Pinch off small pieces and roll with your palms into ropes about 1/8 to 3/16 inch (3–5 mm) thick. Keep thickness even and shorten ropes as needed. Arrange loosely in one layer on a lightly floured board.

Bring a large pot of water to a boil. Meanwhile warm the oil in a skillet over medium-low heat. Add garlic and stir for about 30 seconds, without browning. Add tomatoes and cook for 6–8 minutes, stirring, until softened and juicy.

Boil the fresh pici, stirring gently to separate them. Start checking after 5 minutes; thicker pieces may need 1–3 minutes longer. Check the thickest noodle for a tender center without raw flour. Reserve 1 cup of cooking water before draining.

Transfer pici to the sauce and toss gently. Add reserved water a tablespoon at a time until lightly coated. Remove from heat and fold in Parmesan and basil. Divide into four portions and serve.