Combine the flour, eggs, olive oil, salt, and grated beetroot in a bowl. Knead the mixture until it forms a smooth, elastic dough.
Wrap the dough in plastic wrap and let it rest for about 30 minutes.
In a blender or food processor, blend the roasted beets, goat cheese, minced garlic, salt, and pepper until smooth. Adjust seasoning to taste.
Roll out the pasta dough into thin sheets using a pasta machine or a rolling pin.
Place small dollops of the beet and goat cheese mixture onto one pasta sheet, leaving space between each dollop. Brush the edges with a bit of water to ensure they remain sealed while cooking.
Gently place another pasta sheet on top of the filling. Press down around each mound of filling to seal the ravioli. Use a ravioli cutter or a sharp knife to cut out individual ravioli pieces. Ensure they are properly sealed.
Boil a large pot of salted water. Carefully add the ravioli and cook for 3-4 minutes or until they float to the top.
In a separate pan, melt the butter over medium heat. Add lemon zest and juice. Cook for 1-2 minutes until fragrant.
Drain the cooked ravioli and gently toss them in the lemon butter sauce.
Plate the beet ravioli, drizzle with extra lemon butter sauce, and garnish with dried chives and crumbled goat cheese.