Place the eggs, water, flour and salt in the bowl of the KitchenAid Professional 600 Series 6-Quart Stand Mixer fitted with the flat beater. Mix on speed 2 for 30 seconds. Remove the beater and attach the dough hook. Knead the dough on speed 2 for about 2 minutes. If the dough seems too wet or too dry, add some water or flour in small increments until the dough comes together as desired. Transfer the dough onto a work surface and knead by hand for about 2 minutes, and form it into a ball. Let it rest for 20-30 minutes.
While the pasta dough rests, make the filling. In a bowl, mix together the chèvre, egg, chives, parsley, salt and pepper until well blended. Let it rest in the refrigerator while you finish making the pasta.
Divide the pasta dough into four pieces and keep covered so the dough does not dry out. Attach the KitchenAid Pasta Sheet Roller attachment to the Stand Mixer. Set the adjustment knob on the pasta sheet roller to setting 1 and turn the mixer to speed 2. Flatten one of the dough pieces to about 1 cm in width and feed the dough into the roller. Fold the dough in half and pass it through the roller again. Repeat this kneading process several times until the dough is smooth and pliable, and about the width of the pasta roller, lightly dusting with flour as you work to prevent the dough from sticking. Turn the adjustment knob on the roller to setting 2 and feed the dough through the pasta roller. Change it to setting 3 and pass through again. Continue rolling the pasta until you have reached setting 5 or 6 (depending on how thick you would like your pasta). If necessary, cut the dough in half as it will get long, and this will make it easier to work with the dough. Repeat with remaining pieces of pasta dough until you have long thin sheets.
Place parsley leaves on the pasta, about 2 inches apart, along a sheet of pasta dough. Lightly mist or brush water on the edges of the pasta and lay another sheet of pasta dough on top. With the pasta roller on setting 3, carefully pass it through, pressing the pasta together. Set aside. Repeat with remaining sheets of pasta.
To make the ravioli, place about 1 mounded teaspoon of chèvre filling on top of the parsley laminated pasta. Be careful not to overfill as the pasta will need to seal around the filling. Lightly brush or mist the edges of the pasta with water. Place another sheet of pasta on top, aligning the laminated parsley over the center of filling. Lightly press the pasta together. Use a 2 1/2 inch ravioli cutter to cut the ravioli. Repeat with remaining pasta dough and filling.
Bring a large pot of water to a simmer (avoid a rolling boil - the ravioli could break apart). Season the water with salt and cook the ravioli for about 5 minutes, or until the ravioli floats to the top. Gently retrieve the ravioli with a slotted spoon and transfer to an oiled dish.
While the ravioli is cooking, make the browned butter sauce by melting the butter in a sauté pan over medium heat. Add the garlic and cook the sauce, stirring, for about 5-7 minutes, until the butter is golden brown. Skim any foam from the sauce and discard. Stir in the blanched peas. Remove from heat and add the fresh herbs, and season with salt and pepper.
Drizzle the browned butter sauce over the ravioli. Serve immediately.