Bring a large pot of water to a boil for the pasta.
Heat olive oil in a large skillet with tight-fitting lid over medium-high heat. Add garlic and chili pepper, and swirl 2 minutes. Add tomatoes and shake to coat in oil, season with salt and pepper, and cover pan. Cook until tomatoes burst, about 10-12 minutes, then break up remaining tomatoes with a potato masher and deglaze the sauce with wine. Reduce a minute or two then stir in tarragon and reduce heat to a simmer. Adjust salt and pepper.
When tomatoes begin to burst, season the boiling water with salt then drop pasta and cook 5 minutes; add green beans and cook 3 minutes more. Reserve half a cup of starchy liquid just before draining. Drain pasta and beans, toss with sauce and starchy cooking water for 1-2 minutes.
Serve in shallow bowls topped with lots of coarsely grated ricotta salata cheese.