Preheat the oven to 400 degrees F (200 degrees C).

Combine beets, onion, and garlic cloves in a bowl. Add olive oil, thyme, salt, and pepper; toss to coat. Transfer to a baking sheet.

Roast in the preheated oven until cooked through and tender, about 1 hour.

Remove from the oven and let cool until beets can be handled, 10 to 15 minutes. Remove beet skins.

Transfer mixture to a food processor. Add vegetable broth and lemon juice; pulse until smooth. Transfer to a saucepan.

Stir heavy cream into the beet mixture in the saucepan. Cook over medium-low heat until it begins to simmer, about 5 minutes. Stir in 1/2 cup Parmesan cheese until melted. Set aside.

Fill a large pot with lightly salted water and bring to a rolling boil. Cook fettuccine at a boil until tender yet firm to the bite, about 8 minutes. Drain and return to the pot. Pour beet sauce over pasta and toss until well combined. Top with 1 tablespoon Parmesan cheese and a pinch of additional thyme.