Fill a large pot with water, season generously with salt, and bring it to a boil. While waiting, place a large Dutch oven or a heavy-bottomed pot over medium heat. Pour in the olive oil, and once it's hot, add the shallots. Sauté for a couple of minutes until they turn golden. Stir in the red peppers along with fresh thyme, oregano, and rosemary. Season with salt and pepper.

Let the peppers cook, stirring often, until they become soft and caramelized—this should take about 15 minutes. Add the garlic and a splash of balsamic vinegar, cooking for another minute. Pour in the vegetable stock, bring everything to a simmer, and let it cook over medium-low heat for about 5 minutes.

As the sauce simmers, add the linguine to the boiling water, stirring well. Cook it for one minute less than the package suggests. Remove the herb stems from the sauce, then stir in the cream. Transfer everything to a blender and blend for at least 90 seconds until completely smooth and creamy.

Drain the pasta and return it to the pot. Pour in the sauce along with the grated parmesan, then stir over low heat for about a minute to coat the linguine and melt the cheese. Finally, fold in the parsley.

Serve the pasta with torn burrata on top. Finish with a drizzle of olive oil, extra parmesan, freshly cracked black pepper, and any fresh herbs you like.