Coat the bottom of a large pot with 2 Tbsp. of the olive oil. Add the ground turkey; cook on medium heat. Stir with a wooden spoon to break down the turkey until it is thoroughly cooked, about 5 minutes. Drain any accumulated fat. Remove cooked turkey from the pot and place in a bowl.
Using the same pot, heat the other 2 Tbsp. of olive oil over medium heat. Add the onions. Sprinkle with some salt and pepper; sauté until onions begin to soften, about 2-3 minutes. Add the garlic and zucchini and continue to cook until the zucchini cooks down a bit, about 5-7 minutes.
Add the tomatoes with its juice, tomato and red pepper pastes, chicken broth, dried herbs, and seasonings. Bring to a boil, then reduce heat to medium. Cook about 5-10 minutes. You want to achieve a saucy, not soupy, consistency. Note: If you don't have red pepper paste, use additional tomato paste and add a little paprika.
Return the cooked ground turkey to the pot. Place a cover on the pot; simmer for 15-20 minutes for flavors to blend and sauce to thicken a bit. Stir from time to time.
If sauce seems too thick, add more broth. Adjust seasonings, as needed. **See notes at the bottom.