For the capunti: Dust a sheet pan with semolina and set aside.
Combine the semolina flour and all-purpose flour in a large bowl. Create a well in the center of the flour and pour in 1/2 cup warm water. Mix the dough with a fork, mixing until smooth. If the dough seems dry, add more water to get a smooth, not wet, dough. Let rest for 15 minutes, if time allows.
Working with a quarter of the dough at a time, roll it into pieces about the size of Ping-Pong balls. Roll each piece out to a rope about 12 inches long and 1/2-inch in diameter. Cut the ropes into 1-inch-long pieces. Using the edge of a knife, roll each piece towards you, putting pressure on the dough to create an indentation in the middle so it forms a canoe-like shape. Place the formed capunti on the prepared sheet pan; keep them covered with a damp kitchen towel until ready to cook. Continue until all the dough is used.
Meanwhile, make the cherry tomato sauce: Heat the oil in a large, deep skillet over medium-high heat. Once glistening, add the garlic and cook, stirring, 1 minute. Add cherry tomatoes and half of the basil. Season to taste with salt and pepper. Cover the skillet and cook until the tomatoes burst and break down, about 10 minutes, lowering the heat as needed and stirring occasionally. Turn the heat off; discard the garlic cloves if you like.
Meanwhile, bring a large pot filled two-thirds full of water to a boil. Season with salt. Drop the capunti into the pot and boil until cooked through, 3 to 4 minutes. Drain the pasta, reserving 1 cup of the cooking water. Add the pasta directly to the skillet with the tomato sauce and stir; the pasta will begin to absorb the sauce. If it is looking dry, add a ladle of the starchy reserved pasta water. Continue stirring, adding more water if needed, until you are happy with consistency. Spoon the warm pasta into bowls and top each serving with torn burrata, the remaining basil leaves and freshly grated Parmigiano.