Preheat the oven to 350 degrees F (175 degrees C).

Combine all tomatoes, garlic, chicken broth, olive oil, balsamic vinegar, red pepper flakes, salt, and pepper in a large, glass baking dish; toss to coat.

Roast in the middle of the preheated oven, stirring regularly, until tomatoes are tender, about 45 minutes.

Remove from the oven and stir in basil, parsley, and oregano.

When the tomatoes have been roasting for about 35 minutes, bring a large pot of lightly salted water to a boil.

Cook linguine at a boil until tender yet firm to the bite, 5 to 7 minutes. Drain and return to the pot.

Add the roasted tomatoes and herbs to the linguine and cook over medium-low heat for 2 minutes.

Add ricotta cheese and stir until melted and smooth.