Prepare the vegetables: slice mushrooms, cut zucchini into rounds, remove florets from broccolini, snap ends off asparagus, and prepare snow peas.
In a medium saucepan over medium-high heat, melt the butter. Add cream and Parmigiano Reggiano, stirring until the cheese melts and sauce thickens slightly. Remove from heat.
Bring a large pot of salted water to a boil. Add linguine and cook according to package instructions, minus one minute. Reserve 1/3 cup pasta water before draining.
Add remaining olive oil to the skillet, toss in broccolini, snow peas, and asparagus, and cook for 2 minutes. Then add zucchini, cherry tomatoes, green peas, garlic, salt, and pepper. Cook for another 3 minutes.