Arrange a rack in center of oven; preheat to 400°. Cut spaghetti squash in half lengthwise. Scrape out seeds. Rub inside of squash with oil; season with 1 tsp. salt. Arrange cut side down on a baking sheet.

Roast squash until tender and browned, 35 to 45 minutes. Let cool slightly.

Meanwhile, in a medium heatproof bowl, season broccoli with 1/4 tsp. salt. Pour 1 Tbsp. water into bottom of bowl. Cover and microwave 1 minute. Stir and continue to microwave in 30-second increments until broccoli is bright green and fork-tender, 30 seconds to 1 minute more. Uncover and let cool slightly.

In a large skillet over medium heat, melt butter. Add onion; season with 1/2 tsp. salt. Cook, stirring occasionally, until onion is tender and sweet to the taste, 7 to 8 minutes. Add garlic and cook, stirring, until fragrant, about 1 minute more. Add flour and cook, stirring, until a bubbling paste forms on bottom of skillet, about 1 minute more.

Slowly whisk in milk and remaining 1/4 cup water. Bring to a simmer over medium-high heat and cook, stirring occasionally, until thickened, 1 to 2 minutes. Remove from heat and fold in 3 oz. cheddar until a smooth sauce forms; season with salt, if needed.

Using 2 forks, shred spaghetti squash strands from shell. Measure out 3 cups and transfer to skillet; reserve remaining squash for another use. Add broccoli to skillet and toss to coat in sauce.

Return cheesy spaghetti squash to squash shells. Top with remaining 1 oz. cheddar.

Heat broiler. Broil squash, watching closely, until cheese is melty and blistered on top, 1 to 2 minutes.