Heat oil (2 tablespoons) and butter (2 tablespoons) in a large pot over medium-high heat and saute onion (1 cup) and garlic (1 tablespoon) on medium-low heat until onions are translucent.
Sprinkle flour (1/2 tablespoon), Italian seasoning (2 teaspoons), rosemary (1/2 teaspoon), salt (1/2 teaspoon), and black pepper (1/2 teaspoon) over onions and continue to saute, whisking constantly, until spices are fragrant for about 1 minute.
Add chicken broth (40 ounces) to the pot, stirring to scrape up any brown bits on the bottom of the pot. Add Better Than Bouillon (1 heaping tablespoon), diced tomatoes (1 can), and sugar (1/8 teaspoon), simmer for 20 minutes.
Add the russet potatoes (2) and Parmesan (1 cup) and simmer, stirring often, (so Parmesan doesn't stick on the bottom of the pot) until potatoes are fork-tender for about 20 minutes. Stir in chicken cubes (3 cooked), heavy cream (1 cup), and spinach (4 ounces), and heat through. Adjust seasoning and serve.
Notes
Get the parmesan right. Finely grated parmesan melts seamlessly into the broth—shredded or shaved parmesan won’t melt properly and can leave clumps or sink to the bottom of the pot.
Watch the bottom of the pot. Parmesan loves to stick if the heat is too high. Keep the pot at a steady simmer and stir often during the potato stage.
Adjust consistency at the end. If you want a thicker soup, mash a few potatoes gently against the side of the pot. If you want it thinner, add a splash of warm broth or cream.
Add spinach at the last minute. Overcooking spinach makes it dark and stringy. A quick stir at the end keeps it bright and silky.