Heat a second frying pan over medium-high heat. Add the butter and cook until melted.
Add the flour and whisk in the butter.
Cook for 30 seconds. Pour in ¼ cup of the heavy cream. Whisk until smooth. Remove from heat.
Take 2 garlic cloves and grate over the sauce. Return to heat.
Pour in ¼ cup more of the heavy cream
Whisk until incorporated.
Add juice from 1 lemon and whisk. Remove from heat, grate remaining 2 garlic cloves into the sauce and return to heat.
Pour in remaining ½ cup of cream and whisk well. Cook for 3 minutes. Lower the heat to simmer. Zest the lemon over the sauce and stir in chopped basil, cracked peppercorns, and a pinch of salt. Keep at a simmer.
When the water boils, add the pasta and cook according to the package directions.
Add the white wine to the pan of sautéed vegetables pan and use a spoon to scrape up any browned bits on the pan. Cook at least 30 seconds to let the alcohol burn off, then add the minced garlic and lower the heat. Juice the lemon into the pan. Stir and let simmer.
Drain pasta and add back to the pot. Pour the pan with shallots and mushrooms into the pasta (juice and all).
Toss to coat all the pasta. Set aside the pot with pasta and cover to maintain temperature.