Preheat oven to 400°F. Peel and cube the purple sweet potatoes. Place them on a baking sheet and drizzle the olive oil, salt and pepper on them. Mix together with your hands and bake on 400°F for about 25 minutes, or until roasted and tender.
While the sweet potatoes are baking, prepare the pasta dough. In a stand mixer with the dough hook attachment, mix the flour, eggs, and olive oil on medium speed. Add in the water as needed so the dough is wet but not sticky. Roll out the dough so it is in a rectangle shape, about 3-4 inches wide and 5 inches long. Run the dough through a pasta roller as if you are making a lasagna sheet. You want to run it through a few times until the dough is about 1/16 of an inch thick. If you don't have a pasta roller, you can roll it by hand with a rolling pin, but be aware it is more time consuming and strenuous! (I learned the hard way.) If doing it by hand, make sure the dough is as thin as possible!
Your pasta sheet should be long, thin and about 5-6 inches wide now. Cut your pasta sheet into four sections, each one about 12 inches long, enough to cover the ravioli mold. Place one section over a very well floured ravioli mold—if it is not floured enough, the dough will stick to the mold! Go over it with a rolling pin so the dough is formed to the mold, and the wells have a space for the filling. Cover the dough with a damp towel.
Remove the purple sweet potatoes from the oven and mix them in a food processor with the cheeses, egg and olive oil. Blend until smooth. Return to your ravioli mold and fill each well with about 2 teaspoons of the filling. Take one of the pasta sheets and lay it over the filling. Dab some water on the edges to seal them together, and run the rolling pin over the top of the raviolis to help them completely form and cut through the mold. (Feel free to use your fingers to press down the dough between wells so the jagged ravioli edge cuts through as well.) Gently turn the mold upside down to release the ravioli and peel apart. Use a pizza cutter to separate any raviolis that are still sticking together. Repeat this process with the remaining dough and filling.
In a large pot, bring water to a boil and cook raviolis over medium-high heat until al dente.
While the ravioli is cooking, add the butter, garlic and basil to a skillet and cook until it browns. Once the raviolis are ready, add them to the skillet, along with the radishes. Mix well and add salt and pepper to taste. Serve while warm and enjoy!