Add 2 tablespoons unsalted butter to Instant Pot insert. Let butter melt completely, then add 1 ½ cups holy trinity and stir to incorporate. Sauté vegetables, stirring occasionally, 5 minutes or until tender and fragrant.
When vegetables are tender, add 4 cloves garlic (minced) to Instant Pot. Stir to incorporate, then sauté ingredients 30 to 60 seconds or until garlic is just fragrant.
When garlic is fragrant, add 1 tablespoon Cajun seasoning and 4 cups low-sodium chicken broth. Stir to incorporate, scraping any browned bits of chicken or vegetables off of bottom of insert.
Add 1 14.5-ounce can diced tomatoes and 1 cup uncooked ditalini. Stir to incorporate, then return seared chicken breasts to Instant Pot.
Secure lid on Instant Pot with valve in Sealing position. Set Instant Pot to Manual High Pressure with 4-minute cook time. Instant Pot will pressurize, then cook time will begin.
After 4-minute cook time ends, carefully Quick Release internal pressure. Remove lid from Instant Pot and set aside. Use caution - escaping steam will be very hot.
Transfer chicken breasts to clean cutting board or large bowl. Chop or shred chicken as desired. Set aside.
Set Instant Pot to Sauté mode. Add 1 cup heavy cream and 1 cup grated fresh parmesan. Return chicken to Instant Pot and stir to incorporate. Simmer soup, stirring gently, 3 to 5 minutes or until cream is incorporated and parmesan is melted.
Taste soup and adjust seasoning if desired. Portion soup into serving bowls, garnish with chopped green onions (optional), and serve warm.