Combine cherry tomatoes, oil, garlic, and salt in a saucepan over medium-low heat. Cook and stir until garlic is just toasted, 2 to 3 minutes.
Pour chicken broth into tomato mixture; bring to a simmer and cook until tomatoes start to collapse and burst, about 10 minutes. Stir oregano and red pepper flakes into tomato mixture.
Puree tomato mixture with a stick blender until sauce is smooth.
Bring a large pot of lightly salted water to a boil; add penne and cook, stirring occasionally, until tender yet firm to the bite, about 11 minutes. Drain and return pasta to the pot. Pour tomato sauce over penne and stir until pasta absorbs some of the sauce, 1 to 2 minutes. Stir Parmigiano-Reggiano cheese into pasta and season with salt and pepper to taste.
