Preheat the oven to 375 degrees F (190 degrees C). Individually wrap beets, onion, and head of garlic in aluminum foil. Place on a baking sheet.

Bake in the preheated oven until beets are fork-tender, about 1 hour depending on size. Remove garlic once soft, after about 30 minutes. Allow beets, onion, and garlic to cool to room temperature.

Meanwhile, prepare the dough. In a large bowl, mix together all purpose flour, whole wheat flour, and 1/2 teaspoon salt. Make a well in the center and add in eggs. Gradually add in the water, 1 tablespoon at a time, until dough comes together (more water may be needed depending on how dry your dough is).

Turn dough out onto a work surface and knead until smooth and elastic, about 8 minutes. Divide dough into 4 portions, wrap in plastic, and refrigerate at least 1 hour.

For the filling: Peel skin off beets, onion, and 3 cloves of the roasted garlic. Reserve the rest of the garlic for another use. Cut beets in half, and place them in a food processor. Add onion and garlic cloves; pulse until smooth, scraping down sides as necessary. Add in 1/4 teaspoon salt, 1/4 teaspoon pepper, 1 teaspoon fresh dill, and 4 ounces goat cheese; pulse several times until thoroughly combined.

Remove 1 piece of chilled dough from the fridge. Shape into a rough rectangle about 3x4 inches. Set pasta maker to the widest setting, and run dough through 3 times. Reduce pasta width setting to the next narrowest setting, and run pasta through twice. Continue to run pasta through pasta maker, dropping the width each time, until about 1/16-inch-thick. Cover pasta sheet with a kitchen towel to prevent it from drying out. Repeat with additional pieces of pasta dough, as needed.

Lay one sheet of pasta on your work surface. Scoop teaspoon-sized portions of beet filling onto the dough about 3 inches apart. Lightly brush water around the edges of the filling. Gently place another sheet of pasta over the filling and press pieces together around the filling, removing as much air as possible. Seal firmly.

Using a 2 1/2-inch cutter, cut raviolis from the dough, ensuring the filling is as centered as possible. Gently press edges again to seal. Place finished ravioli on a lightly floured baking sheet. Continue making pasta and filling raviolis until all the beet filling is used. Reserve any extra dough for another use.

When ready to serve, bring a large pot of salted water to a boil. Add a few ravioli at a time to the boiling water. Boil until they rise to the top, 3 to 5 minutes. Remove from water with a slotted spoon and set aside.

Melt 2 tablespoons of unsalted butter in a second saucepan over medium heat. Add a few ravioli at a time to the pan and cook for 2 minutes. Flip over and cook an additional 2 minutes.