Noodles with Chilled Tomato Broth

Name: Noodles with Chilled Tomato Broth
Ingredients: 1½ lb. very ripe red tomatoes, chopped; 1 small garlic clove, smashed; 3 Tbsp. unseasoned rice vinegar; 1 Tbsp. sugar; 1 Tbsp. plus 1 tsp. kosher salt, plus more; 10 oz. thin noodles (preferably somen); 5 large eggs, beaten to blend; ½ Tbsp. unsalted butter; 2 medium Persian cucumbers, halved, thinly sliced on a diagonal; 3 scallions, thinly sliced; ½ cup cilantro leaves with tender stems; Hot chili paste (such as sambal oelek; for serving)