Season tomatoes and shallot with salt. Toss with chile, vinegar and 1/4 cup oil. Let stand for at least 15 minutes or up to 1 hour.
Bring a large pot of salted water to a boil. Fill a large bowl with ice and water. Boil wax beans until crisp-tender, about 5 minutes. Drain, transfer to ice water and let cool to room temperature. Drain again. Cut beans into 2-inch lengths at an angle.
Combine goat cheese, lime juice, remaining 1 Tbsp. oil, 2 Tbsp. water and a pinch of salt in a blender. Puree until smooth. Toss with beans until well coated.
Place wax beans in 6 serving dishes. Transfer tomato mixture with a little marinade to beans. Season salad with black pepper, tear basil leaves on top and serve.