Herbed Fresh Pasta with Market Tomatoes and Peppers

Combine 1 cup plus 2 Tbsp. semolina flour and 1 cup all-purpose flour in a large bowl. Purée eggs, basil, and 1 tsp. salt in a blender on medium-high speed, scraping down sides as necessary, until mixture is smooth, about 30 seconds. Scrape basil mixture into flour mixture and stir with a fork until dough just comes together. Turn out onto a work surface lightly dusted with all-purpose flour and knead until smooth and elastic, 8–10 minutes (alternatively, use a stand mixer fitted with a dough hook and beat on medium-low speed, about 8 minutes). Shape into a 6" log and wrap tightly in plastic wrap; let sit 1 hour at room temperature.

Cut dough into 6 equal pieces. Working with one piece at a time, lightly dust both sides with all-purpose flour and roll into a thin 5½x20" rectangle (you should be able to see the shadow of your hand pass behind the sheet of pasta; alternatively, use a pasta machine). Dust both sides of dough with semolina flour and loosely roll into a cylinder. Using a sharp knife, trim edges and discard. Cut dough into 1½"-wide slices. Unfurl noodles, dust with semolina flour, and gently toss to separate. Place on a rimmed baking sheet and cover with a clean kitchen towel.

Do Ahead: Pasta dough can be made 1 day ahead. Wrap tightly and chill.

Preheat oven to 425°. Toss plum tomatoes, peppers, 2 garlic cloves, and ¼ cup oil on a rimmed baking sheet. Arrange tomatoes and peppers in a single layer, skin side down; season with salt and pepper. Roast until browned around the edges, 35–45 minutes. Let sit until cool enough to handle, then slip off skins. Transfer tomatoes and peppers to a large bowl and mash into medium-size pieces using a potato masher. Set aside.

Combine cherry tomatoes, 2 Tbsp. oil, and remaining garlic clove in a small bowl. Season with salt and pepper and let sit at room temperature 30 minutes.

Meanwhile, cook pasta in a large pot of boiling salted water, stirring occasionally, until al dente, about 4 minutes. Drain, reserving 1 cup pasta cooking liquid.

Toss pasta with tomato and pepper mixture, adding ⅓ cup pasta cooking liquid and remaining 2 Tbsp. oil to keep pasta from sticking; add more pasta cooking liquid if necessary. Divide pasta among plates, then top with cherry tomato mixture and any accumulated juices, basil, and Parmesan.