Place 3 quarts of water in a large saucepan over high heat and bring to a boil. Add the broccoli florets and lower the heat until the water is simmering.
Cook for about 5 minutes. Using tongs, remove the florets and place them on a paper towel to drain. Leave the water in the saucepan.
While the broccoli is cooking, heat the olive oil in a 10-inch non-stick skillet over medium heat and add the shrimp. Cook for about 3 minutes on each side and transfer to a plate.
Add the minced garlic to the skillet. Cook very slowing and stir frequently. Do not allow the garlic to brown or it will become bitter.
Add the flour slowly and cook for about one minute. Stir continuously to blend the oil and flour. The mixture will be like coarse cornmeal. Cook gently so the mixture does not brown.
Slowly add the cold milk, whisking to keep the sauce from forming clumps. Blend all the milk until the sauce is smooth and begins to thicken.
Add the goat cheese and whisk as it melts. When the sauce is smooth, add the Parmigiano-Reggiano and whisk as it melts until the sauce is creamy. Lower the heat to very low.
Add 2 cups of water to the saucepan you cooked the broccoli in and heat the water to a boil. Add the fettuccine and cook until just tender (10-12 minutes for dried pasta).
When the pasta is almost done, add the shrimp and broccoli to the Alfredo sauce and toss to coat well. Increase the heat to medium.
Drain the pasta well and then add it to the sauce, tossing to coat thoroughly.