In a large pot of boiling, salted water, cook the pasta until al dente, about 10 minutes. Drain, reserving 1 cup of the pasta cooking water.
Meanwhile, in a medium bowl, mix the yogurt and garlic; season with salt and the cayenne.
In a large nonstick skillet, cook the sausage over medium-high heat, stirring occasionally, until cooked through, 6 to 8 minutes. Add the peas and season with salt and black pepper. Cook until just heated through. Add the pasta and 1/2 cup of the reserved pasta water and toss, adding the remaining pasta water if needed to thin the sauce; season with salt and black pepper.
Divide the pasta among 4 bowls. Top with the yogurt mixture and mint.