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1
Place carrots in a large pot with lightly salted water to cover. Bring to a boil and cook carrots until fork-tender, 7 to 10 minutes. Drain.
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2
Meanwhile, melt butter in a saucepan over medium heat. Add flour and mix well. Add heavy cream and bring to a simmer over medium heat. Cook and stir until a thick roux forms, about 10 minutes.
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3
Mix fontina cheese and goat cheese into the roux. Cook until cheeses are completely melted and sauce is thickened, 3 to 5 minutes. Remove from heat.
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4
Drain carrots and bring water back to a boil, pouring in more as needed. Cook spaghetti in the boiling water, stirring occasionally, until tender yet firm to the bite, about 12 minutes.
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5
Cook garlic with olive oil in another saucepan over medium heat until starting to brown, 1 to 2 minutes. Add Swiss chard, tearing them as they go into the pan. Cook until wilted, about 2 minutes. Add baby arugula and cook until wilted, about 1 minute. Chop 1 basil leaf into the pan using herb shears. Cook until wilted, about 30 seconds more. Mix in some of the cheese sauce if desired.
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6
Drain pasta and divide over serving plates. Add carrots, chard, and arugula. Top with Parmesan cheese and more cheese sauce. Garnish each plate with a basil leaf.
