Bring a large pot of salted water to a boil. Meanwhile, combine 1 tablespoon water and saffron in a small bowl; set aside.
In a large nonstick skillet, heat oil over medium-high heat. Add onion and thyme; cook, stirring frequently, until onion is softened, about 4 minutes, then add sausage and cook, breaking up sausage into small bits with a wooden spoon, until sausage is mostly cooked through, about 6 minutes. Add cream, saffron mixture and generous pinch salt; bring to a simmer and cook for 2 to 3 minutes. Remove from heat and cover to keep warm.
Add pasta to the boiling water and cook until al dente. Drain, then transfer to a large bowl; immediately add cream mixture and toss to combine. Adjust seasoning and serve immediately.