Bring 1 to 2 inches of water to a boil in a large saucepan with a steamer basket. Steam the broccoli until very tender and bright green, about 5 to 6 minutes. Then remove the steamer basket and discard the water.
In a small skillet, combine olive oil and garlic. Cook over medium-high heat, stirring frequently, until the garlic is golden brown, approximately 2 to 3 minutes. Remove from heat and carefully pour in the vegetable broth to stop the garlic from cooking further.
Transfer half of the steamed broccoli to a food processor. Add the sautéed garlic mixture, salt, thyme, and white pepper. Puree until smooth and transfer back to the saucepan.
Pulse the remaining broccoli in the food processor to your preferred texture. Combine it with the pureed broccoli in the saucepan, stirring well. Serve the Garlic Mashed Broccoli hot and enjoy.