For Sauce:
In a large heavy bottom pot, add 2 Tbsp. of olive oil over medium heat. Sauté minced garlic for about 30 seconds to 1 minute, taking care not to brown. Add ground beef, ground pork, salt and pepper. Using a spatula break up meat into small pieces as it cooks.
When cooked through, add tomatoes, paste, water and balsamic vinegar. Stir well and simmer over medium/low heat for at least 1 hour. Add grated parmesan and cook for another 10 minutes. If you have time, cook all day over low heat, stirring occasionally, for a deeper richer flavor. Can be made in advance and refrigerated. When ready to use, heat on stove. Or make a double recipe and freeze what you are not using in air-tight container for up to 3 months.
For Gnocchi:
Bring a pot of water to a boil and add peeled potatoes. Boil until soft, but not mushy, about 20 minutes depending upon size of potatoes. Test by pricking with a fork through to center. Drain While still hot, peel potatoes by holding with a pot holder and peeling off skins.
Use a ricer or the smaller holes of a box grater to grate potatoes into a large bowl. Add egg yolks, ricotta and parmesan cheese and salt. Stir to combine. Sprinkle half the flour into mix and gently stir to incorporate. Add remaining flour and repeat. Dough should be loosely coming together.
On a lightly floured surface gently knead dough and form a ball.
Divide balls into 4 equal pieces.
Take one piece and roll into a long log about ¾ inch in diameter. Using a sharp knife or bench scraper, cut log into ¾ inch pieces. Using a grooved pasta board, or fork, gently roll each piece down board with your thumb to curl and shape. Set rolled pasta on a lightly floured, parchment lined baking sheet in a single layer. Repeat for each additional piece of dough.
Set pasta aside with a slightly damp clean kitchen towel over the gnocchi.
Preheat oven to 400°F.
In a cast iron skillet, or other oven-proof pan/dish, heat 1 Tbsp olive oil. Add onions and sauté over medium heat until slightly translucent. Scoop a generous ladleful of the meat sauce into the pan. Add one layer of the gnocchi. Add more sauce and then break apart one 4-ounce ball of burrata over sauce. Add a second layer in the same manner. Sprinkle parmesan over the top.
Bake in oven for 15 minutes, until sauce is bubbly and hot. Finish off, by broiling on high for about 1-2 minutes, watching careful so as not to burn cheese.
Remove from oven and serve.