Warm oil in a nonstick skillet over medium-low heat. Add shallot; cook, stirring occasionally, 1 minute. Add chard and 1/4 teaspoon pepper; cook 3 minutes or until leaves are wilted, stirring occasionally. Transfer chard to a bowl, set aside, and keep warm. In another bowl, combine ricotta, goat cheese, Parmesan, chopped sage, and remaining 1/4 teaspoon pepper.
Lay out half of the wonton wrappers on a parchment paper–lined baking sheet. Working with 1 wonton at a time, spoon 2 teaspoons of the cheese mixture into center of wrapper. Dip your fingertip in water, and wet edges of wrapper.
Lay another wrapper over mound of filling, and press gently to seal edges. Repeat with remaining wrappers and filling. (At this point, you can cover ravioli with plastic wrap and refrigerate up to 1 day.)
Bring a large pot of salted water to a boil; reduce to a simmer. Drop ravioli into water, stirring gently to prevent sticking. Cook 5 minutes. Remove ravioli with a slotted spoon; place on paper towels to blot.
Arrange 6 ravioli in each of 6 shallow bowls. Drizzle 1 tablespoon pesto over each bowl of pasta. Divide cooked chard over each serving. Garnish with sage, if desired.