For the pasta, whisk together eggs, 2 tbsp. oil, and vinegar in a small bowl. Place flour in a large mixing bowl or food processor. Gradually work egg mixture into flour, then turn out onto a lightly floured surface and knead until smooth and elastic, about 5 minutes. Form dough into a ball, wrap in plastic, and refrigerate for 1 hour.
For the filling, heat 4 tbsp. oil in a large skillet over medium heat. Add onions and cook, stirring frequently, until they begin to soften, about 7 minutes. Add borage, garlic, basil, sage, parsley, and thyme and season with salt and pepper, then cook, stirring frequently, until greens wilt, about 5 minutes. Drain and allow to cool. Place borage mixture in a clean dish towel and squeeze dry. Transfer to a large bowl. Mix in cheese and adjust seasoning with salt and pepper.
Divide dough into four parts. Using a pasta machine, roll out dough as thinly as possible, then transfer to a lightly floured surface. Cut out about 60 rounds using a 2'' cutter. To assemble ravioli, spoon about 1 tsp. of filling into center of a dough round, brush edge with water, then top with another round, pinching to seal. Repeat process, using up all pasta rounds and filling.
For the sauce, bring stock and demi-glace to a simmer in a medium saucepan over medium heat. Reduce by three-quarters, 20–30 minutes, then whisk in butter. Meanwhile, cook ravioli in batches in a large pot of boiling salted water until they rise to the surface, 3–4 minutes. Drain and divide between four small plates. Top with sauce and garnish with julienned sage.