Pat beef chuck roast cubes dry and season all sides with salt and pepper. Set instant pot to SAUTE mode and heat olive oil until shimmery. Brown chuck roast cubes on all sides until golden brown.
Add vegetables, rosemary, thyme, wine, beef broth, and tomatoes to instant pot and seal lid. Cook on high pressure 30 minutes then let release pressure naturally for 5 minutes.
Remove herb sprigs from sauce. Using two forks, shred beef into medium pieces and return to pot. Remove 1/2 cup liquid from instant pot and gradually whisk in flour until smooth. Add back to instant pot and stir until sauce is smooth and thickened.
Add butter to pot and stir until butter is melted. Season sauce with salt and pepper to taste and keep warm until ready to serve.
For serving, spoon hot beef ragu over cooked pappardelle pasta and top with grated Parmesan. Enjoy!