Bring a large pot of generously salted (use about 1/2 cup kosher salt) water to a boil. Set up an ice bath next to the stove and generously salt (about 1/4 cup kosher salt) that water as well.

Cut the broccoli stems from the heads; slice the florets into small pieces, each about 1-inch long. Peel the stems with a peeler and cut each stem in half lengthwise, then slice into 1/2-inch-thick half-moons. Add broccoli to the boiling water and blanch for 30 seconds; transfer to the ice water to shock the broccoli and stop the cooking. Drain broccoli in a colander, then set aside (still in the colander) to drain completely.

In a large dry skillet, toast pecans over medium heat until they smell nutty, about 1 minute; remove pecans and set aside.

Return skillet to medium heat and add bacon and oil. Cook, stirring occasionally, until bacon has rendered its fat and is crispy, about 10 minutes.

Turn heat to low and whisk in mustard and vinegar. Add green onions and pecans and season generously with black pepper.

In a large bowl, combine broccoli and grapes. Pour the contents of the skillet into the bowl and toss to combine; season to taste with salt and pepper. Sprinkle with the cheddar and toss right before serving.